#7312819

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7220015);