#6467879

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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)(15,20)(16,21)(17,19)(18,22)(23,34)(24,33)(25,29)(26,36)(27,31)(28,30)(32,38)(35,40)(37,43)(39,41)(42,44);(1,2,3,4)(14,19)(15,23)(16,26)(18,27)(20,32)(21,35)(24,39)(25,38)(28,40)(29,34)(30,36)(42,43);(6,9)(10,12)(13,15,18,23,17,25,27,38)(14,20,22,34,19,29,31,32)(16,24,26,37,28,39,40,44)(21,33,36,43,30,41,35,42);(13,16,27,40,17,28,18,26)(14,21,31,35,19,30,22,36)(15,24,38,44,25,39,23,37)(20,33,32,42,29,41,34,43);(5,7,8,11)(6,10,9,12)(15,25)(20,29)(23,38)(24,39)(32,34)(33,41)(37,44)(42,43)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=hi,e^2=g,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, 6375075);