#72287836

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623513	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,7)(4,8)(10,11)(12,15);(1,2,5,10,6,7,13,11)(3,12,14,8,9,15,16,4)(17,18)(19,29,32,25)(20,34,37,24)(21,38)(22,40)(23,28)(26,36)(27,35)(30,46)(31,43,48,44)(33,45,47,42)(39,41);(1,3,13,16,6,9,5,14)(2,8,10,15,7,4,11,12)(17,19,30,37,22,33,39,48,23,32,21,20,35,47,36,31)(18,24,41,25,27,43,46,45,28,34,26,29,40,44,38,42);(1,3,5,14,6,9,13,16)(2,8,11,12,7,4,10,15)(17,20,36,32,35,48,21,33,23,37,39,19,22,31,30,47)(18,25,38,24,40,42,26,44,28,29,46,34,27,45,41,43);(1,4)(2,9)(3,7)(5,12)(6,8)(10,16)(11,14)(13,15)(19,31)(20,33)(21,39)(22,35)(24,42)(25,44)(26,46)(27,40)(29,43)(30,36)(32,48)(34,45)(37,47)(38,41)

Polycyclic presentation:

2^10|b^2=f,c^2=ghj,d^2=ghi,f^2=h,g^2=i,i^2=j,b^a=bh,c^b=cg,d^b=dghij,d^c=dhj,e^a=eh,e^b=eh,e^c=eghj,e^d=eghij,f^c=fj,f^d=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318603, #72318841, #72319621
maximal normal subgroup of	#72318603, #72318841, #72319621

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);