#7091641

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 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)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,21)(22,25)(23,26)(24,28)(29,32)(30,31);(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,26,32)(20,28,22,29)(21,31,25,27);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,22)(18,25,26,21)(19,28,30,29)(24,27,32,31);(11,16)(13,15)(17,21,23,25)(18,20,26,22)(19,24,30,32)(27,28,31,29);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,20,23,22)(18,21,26,25)(19,29,30,28)(24,27,32,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,f^d=fi

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, 6998837);