#6630192

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2149 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,c^b=ci,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537388);