#8095843

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

4887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003039);