#6627733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534929);