#7309087

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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)(3,7)(4,6)(5,8)(17,18,24,32,23,20,22,27)(19,25,21,31,28,30,29,26);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,28)(18,25,20,30)(21,22,29,24)(26,32,31,27);(9,11)(10,13)(12,16)(14,15)(17,20,23,18)(19,25,28,30)(21,26,29,31)(22,27,24,32);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,22)(20,31)(23,29)(24,28)(25,32)(27,30);(2,6)(5,7)(10,14)(13,15)(19,28)(21,29)(25,30)(26,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,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, 7216283);