#6514256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499814
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

1799 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)(13,14,19,24)(15,20,26,34)(16,29,30,38)(17,23,18,22)(21,32,37,31)(25,39,41,44)(27,36,28,35)(33,42,43,40);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,35)(23,36)(24,34)(29,39)(30,41)(31,40)(32,42)(37,43)(38,44);(1,3,4,2)(6,9)(7,11)(13,16)(14,21)(15,25)(17,31)(18,32)(19,30)(20,33)(22,29)(23,38)(24,37)(26,41)(27,40)(28,42)(34,43)(35,39)(36,44);(5,7,8,11)(6,10,9,12)(15,26)(16,30)(20,34)(21,37)(27,28)(29,38)(31,32)(35,36);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,27,26,28)(16,31,30,32)(20,35,34,36)(21,29,37,38)(25,40,41,42)(33,39,43,44)

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,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421452);