#8429634

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

5072 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8336830);