#6554699

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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)(13,14,17,21,18,22,32,38)(15,23,27,34,26,19,25,35)(16,28,31,37,30,36,29,20)(24,40,41,33,42,44,43,39);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,15,18,26)(14,19,22,23)(16,24,30,42)(17,27,32,25)(20,33,37,39)(21,35,38,34)(28,44,36,40)(29,43,31,41);(1,3,4,2)(5,6,10,9)(7,12,8,11)(13,16)(14,20)(15,24)(17,29)(18,30)(19,33)(21,36)(22,37)(23,39)(25,41)(26,42)(27,43)(28,38)(31,32)(34,44)(35,40);(5,8,10,7)(6,11,9,12)(13,17,18,32)(14,21,22,38)(15,25,26,27)(16,29,30,31)(19,34,23,35)(20,36,37,28)(24,41,42,43)(33,44,39,40);(5,9,10,6)(7,12,8,11)(14,22)(19,23)(20,37)(21,38)(28,36)(33,39)(34,35)(40,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fi,d^2=hi,e^2=h,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, 6461895);