#6466016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

1974 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)(16,27)(20,35)(24,39)(28,42)(30,36)(33,40)(37,44)(41,43);(2,4)(13,14,18,22)(15,23,25,38)(16,28,27,42)(17,31,29,21)(19,32,34,26)(20,36,35,30)(24,40,39,33)(37,43,44,41);(5,7,8,12)(6,9,10,11)(13,15,17,26,18,25,29,32)(14,19,21,23,22,34,31,38)(16,24,30,41,27,39,36,43)(20,33,28,37,35,40,42,44);(13,16,29,36,18,27,17,30)(14,20,31,42,22,35,21,28)(15,24,32,43,25,39,26,41)(19,33,38,44,34,40,23,37);(15,25)(19,34)(23,38)(24,39)(26,32)(33,40)(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,d^a=di,c^b=ch,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373212);