#72319357

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	#7623097	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2324 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:

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

Polycyclic presentation:

2^10|b^2=f,d^2=fi,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=cj,c^b=cg,d^b=dhij,d^c=dh,e^b=eh,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319381, #72319579, #72319699
maximal normal subgroup of	#72319381, #72319579, #72319699

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, 4, F.6*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, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);