#7409609

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

1662 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)(19,25)(20,31)(21,27)(22,28)(23,32)(24,29)(26,30);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,24,30)(18,25,29,26)(20,22,21,23)(27,32,31,28);(10,12)(11,15)(17,20)(18,26)(19,23)(21,24)(22,30)(25,29)(27,28)(31,32);(1,3,5,4)(2,6,7,8)(9,12,13,10)(11,14,15,16)(17,21,24,20)(18,27,29,31)(19,22,30,23)(25,28,26,32);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(17,22,24,23)(18,28,29,32)(19,20,30,21)(25,31,26,27)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7316805);