#6467883

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

1832 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)(16,26)(17,19)(18,22)(21,28)(23,34)(24,39)(25,30)(27,35)(29,31)(32,38)(33,40)(36,41)(37,42)(43,44);(1,2,3,4)(14,19)(15,23)(16,27)(18,29)(20,32)(21,35)(24,40)(25,38)(26,36)(28,41)(30,34)(42,43);(6,9)(10,12)(13,15,18,23,17,25,29,38)(14,20,22,34,19,30,31,32)(16,24,27,37,28,40,41,44)(21,33,36,43,26,39,35,42);(13,16,18,27,17,28,29,41)(14,21,22,36,19,26,31,35)(15,24,23,37,25,40,38,44)(20,33,34,43,30,39,32,42);(5,7,8,11)(6,10,9,12)(15,25)(20,30)(23,38)(24,40)(32,34)(33,39)(37,44)(42,43)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,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,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, 6375079);