#8716824

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

4986 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)(3,8)(4,5)(6,7)(17,18,23,22)(19,26,21,24)(20,28,29,25)(27,31,30,32);(1,3,6,5)(2,4,7,8)(9,10,13,16)(11,12,15,14)(17,19,23,21)(18,24,22,26)(20,27,29,30)(25,31,28,32);(1,3,6,5)(2,4,7,8)(9,11)(10,14)(12,16)(13,15)(20,29)(25,28)(27,30)(31,32);(2,7)(3,5)(9,12,13,14)(10,15,16,11)(17,20,23,29)(18,25,22,28)(19,27,21,30)(24,31,26,32);(1,4,6,8)(2,3,7,5)(9,11)(10,12)(13,15)(14,16)(17,21,23,19)(18,24,22,26)(20,30,29,27)(25,31,28,32);(1,5,6,3)(2,4,7,8)(9,12)(10,15)(11,16)(13,14)(17,22,23,18)(19,26,21,24)(20,25,29,28)(27,31,30,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,e^2=g,f^2=g,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, 8624020);