#9124703

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

4762 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9031899);