#9070557

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

4994 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,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, 8977753);