#8527477

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

4747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8434673);