#6514180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 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)(13,14,18,22)(15,19,24,33)(16,26,27,36)(17,29,31,21)(20,30,35,28)(23,37,39,44)(25,41,40,34)(32,42,43,38);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,22,33)(16,23,27,39)(17,25,31,40)(20,32,35,43)(21,34,29,41)(26,37,36,44)(28,38,30,42);(1,3,4,2)(6,9)(7,11)(13,16)(14,20)(15,23)(17,30)(18,27)(19,32)(21,36)(22,35)(24,39)(25,42)(26,29)(28,31)(33,43)(34,44)(37,41)(38,40);(5,7,8,11)(6,10,9,12)(15,24)(16,27)(19,33)(20,35)(25,40)(26,36)(28,30)(34,41);(6,9)(10,12)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421376);