#6467099

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1996 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)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,26)(17,21)(18,22)(20,29)(23,39)(24,33)(25,34)(27,42)(28,36)(30,43)(31,38)(32,41)(35,44)(37,40);(2,4)(13,15,17,24)(14,19,21,33)(16,27,29,44)(18,31,28,25)(20,35,26,42)(22,38,36,34)(23,40,41,30)(32,43,39,37);(6,9)(10,12)(13,16,28,40,17,29,18,30)(14,20,36,43,21,26,22,37)(15,23,31,44,24,41,25,27)(19,32,38,42,33,39,34,35);(13,16,28,40,17,29,18,30)(14,20,36,43,21,26,22,37)(15,23,31,44,24,41,25,27)(19,32,38,42,33,39,34,35);(5,7,8,11)(6,10,9,12)(14,21)(19,33)(20,26)(22,36)(32,39)(34,38)(35,42)(37,43)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,h^b=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, 6374295);