#8288383

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8195579);