#6554696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461892);