#72287142

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	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2330 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=g,d^2=f,f^2=h,g^2=i,i^2=j,c^b=chj,d^a=dh,d^c=dghi,e^c=eg,f^c=fj,g^d=gi,g^e=gij,i^d=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287726, #72318591
maximal normal subgroup of	#72287278, #72287726, #72318591

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