#6467084

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,31)(24,33)(25,32)(26,34)(27,35)(28,37)(29,36)(30,38)(39,42)(40,43)(41,44);(2,4)(13,15,18,25)(14,19,21,32)(16,26,29,41)(17,30,27,24)(20,34,36,44)(22,38,35,33)(23,39,40,28)(31,42,43,37);(6,9)(10,12)(13,16,27,39,18,29,17,28)(14,20,35,42,21,36,22,37)(15,23,30,41,25,40,24,26)(19,31,38,44,32,43,33,34);(13,16,27,39,18,29,17,28)(14,20,35,42,21,36,22,37)(15,23,30,41,25,40,24,26)(19,31,38,44,32,43,33,34);(5,7,8,11)(6,10,9,12)(14,21)(19,32)(20,36)(22,35)(31,43)(33,38)(34,44)(37,42)

Polycyclic presentation:

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

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, 6374280);