#6574371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481567);