#9123541

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
prime radical	#59074	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4664 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,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, 9030737);