#8704107

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

4738 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,f^2=h,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, 8611303);