#9123794

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

4893 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^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, 9030990);