#6467516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2300 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)(13,14,17,22,18,23,26,19)(15,24,25,31,28,20,33,35)(16,21,29,36,30,37,40,32)(27,38,39,42,41,34,43,44);(2,4)(14,19)(15,25)(16,29)(17,26)(22,23)(27,41)(28,33)(30,40)(31,35)(32,36)(34,44)(38,42);(6,9)(10,12)(13,15,26,33,18,28,17,25)(14,20,19,31,23,24,22,35)(16,27,40,43,30,41,29,39)(21,34,32,42,37,38,36,44);(13,16,17,29,18,30,26,40)(14,21,22,36,23,37,19,32)(15,27,25,39,28,41,33,43)(20,34,35,44,24,38,31,42);(5,7,8,11)(6,10,9,12)(15,28)(20,24)(25,33)(27,41)(31,35)(34,38)(39,43)(42,44)

Polycyclic presentation:

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

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, 6374712);