#6467912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

1843 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)(16,27)(17,19)(18,22)(21,30)(23,35)(24,32)(25,37)(26,31)(28,36)(29,34)(33,40)(38,41)(39,43)(42,44);(1,2,3,4)(14,19)(15,23)(16,28)(18,24)(20,33)(21,36)(25,29)(26,40)(27,38)(30,41)(31,35)(43,44);(6,9)(10,12)(13,15,24,40,17,26,18,23)(14,20,32,33,19,31,22,35)(16,29,41,39,30,25,28,42)(21,37,36,44,27,34,38,43);(13,16,18,28,17,30,24,41)(14,21,22,38,19,27,32,36)(15,25,23,39,26,29,40,42)(20,34,35,44,31,37,33,43);(5,7,8,11)(6,10,9,12)(16,30)(21,27)(25,29)(28,41)(34,37)(36,38)(39,42)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375108);