#9137635

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

4886 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=h,f^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, 9044831);