#9210521

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

4719 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=g,f^2=g,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, 9117717);