#8657341

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

4999 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8564537);