#6514183

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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)(13,14,19,23)(15,20,25,34)(16,28,29,37)(17,22,18,31)(21,32,36,30)(24,38,40,43)(26,35,27,41)(33,42,44,39);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,33)(22,35)(23,34)(28,38)(29,40)(30,39)(31,41)(32,42)(36,44)(37,43);(1,3,4,2)(6,9)(7,11)(13,16)(14,21)(15,24)(17,30)(18,32)(19,29)(20,33)(22,37)(23,36)(25,40)(26,39)(27,42)(28,31)(34,44)(35,43)(38,41);(5,7,8,11)(6,10,9,12)(15,25)(16,29)(20,34)(21,36)(26,27)(28,37)(30,32)(35,41);(6,9)(10,12)(13,17,19,18)(14,22,23,31)(15,26,25,27)(16,30,29,32)(20,35,34,41)(21,37,36,28)(24,39,40,42)(33,43,44,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=h,e^2=i,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, 6421379);