#6484968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(2,5)(6,7)(10,14)(11,12)(17,19,22,18,24,26,23,27)(20,30,32,29,21,28,31,25);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,24)(22,32)(23,31)(26,30)(27,29);(9,12,13,11)(10,15,14,16)(17,21,24,20)(18,25,27,29)(19,28,26,30)(22,31,23,32);(10,14)(15,16)(17,22,24,23)(18,26,27,19)(20,31,21,32)(25,28,29,30)

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,d^2=h,e^2=i,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, 6392164);