#7409784

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7316980);