#6456352

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	#25087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1933 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363548);