#8106068

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

4925 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013264);