#7017672

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1287 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)(17,18,23,24,22,20,19,26)(21,25,30,31,29,28,27,32);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,20,26)(21,27,29,30)(25,31,28,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,18)(19,26,23,24)(21,28,29,25)(27,32,30,31);(1,3,5,4)(2,8,6,7)(11,16)(13,15)(17,21,22,29)(18,25,20,28)(19,27,23,30)(24,31,26,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(21,29)(25,28)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,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, 6924868);