#8726601

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

4803 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8633797);