#6668602

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575798);