#7009548

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6916744);