#6479799

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,25)(22,31)(23,26)(24,29)(27,28)(30,32);(11,12)(14,16)(17,19,22,25,23,24,28,32)(18,21,27,20,26,30,31,29);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,23,29)(18,24,26,19)(21,22,30,28)(25,27,32,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,21,23,30)(18,25,26,32)(19,27,24,31)(20,28,29,22);(10,15)(14,16)(18,26)(20,29)(21,30)(27,31)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6386995);