#9070549

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

4778 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,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, 8977745);