#7221672

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1213 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7128868);