#6567600

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 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)(15,20)(16,29)(17,22)(18,23)(19,24)(21,30)(25,39)(26,34)(27,35)(28,36)(31,37)(32,38)(33,41)(40,43)(42,44);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,15,19,28)(14,20,24,36)(16,25,31,42)(17,27,18,26)(21,33,38,43)(22,35,23,34)(29,39,37,44)(30,41,32,40);(1,3,4,2)(5,6,10,9)(7,12,8,11)(13,16,19,31)(14,21,24,38)(15,25,28,42)(17,30,18,32)(20,33,36,43)(22,37,23,29)(26,40,27,41)(34,39,35,44);(5,8,10,7)(6,11,9,12)(13,17,19,18)(14,22,24,23)(15,26,28,27)(16,30,31,32)(20,34,36,35)(21,37,38,29)(25,40,42,41)(33,39,43,44);(5,9,10,6)(7,12,8,11)(13,17,19,18)(14,23,24,22)(15,27,28,26)(16,30,31,32)(20,34,36,35)(21,29,38,37)(25,41,42,40)(33,39,43,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fi,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,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, 6474796);