#9130357

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

4895 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,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, 9037553);