#9224273

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

5111 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,e^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, 9131469);