#9229771

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

5292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9136967);