#6630271

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2158 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)(13,14,18,22)(15,19,24,32)(16,26,29,41)(17,30,27,21)(20,34,35,28)(23,36,38,44)(25,40,37,33)(31,42,43,39);(6,9)(10,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,33)(22,32)(26,36)(27,37)(28,39)(29,38)(30,40)(34,42)(35,43)(41,44);(13,16,27,34,18,29,17,28)(14,20,30,41,22,35,21,26)(15,23,37,42,24,38,25,39)(19,31,40,44,32,43,33,36);(2,4)(15,24)(19,32)(23,38)(25,37)(31,43)(33,40)(36,44)(39,42);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di

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, 6537467);