#7086660

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,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, 6993856);