#6466011

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ 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)(5,6,8,10)(7,11,12,9);(2,4)(13,14,18,22)(15,23,25,38)(16,27,30,42)(17,31,28,21)(19,32,34,26)(20,35,36,29)(24,39,40,33)(37,43,44,41);(5,7,8,12)(6,9,10,11)(13,15,17,26,18,25,28,32)(14,19,21,23,22,34,31,38)(16,24,29,41,30,40,35,43)(20,33,27,37,36,39,42,44);(13,16,28,35,18,30,17,29)(14,20,31,42,22,36,21,27)(15,24,32,43,25,40,26,41)(19,33,38,44,34,39,23,37);(15,25)(19,34)(23,38)(24,40)(26,32)(33,39)(37,44)(41,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6373207);