#9161103

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	#43105	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4748 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)(17,18,23,28)(19,26,30,24)(20,31,27,22)(21,29,32,25);(3,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,23,30)(18,24,28,26)(20,21,27,32)(22,29,31,25);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,20,23,27)(18,22,28,31)(19,21,30,32)(24,29,26,25);(11,14)(12,16)(17,21,23,32)(18,25,28,29)(19,27,30,20)(22,26,31,24);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,15,11)(17,19)(18,26)(20,21)(22,25)(23,30)(24,28)(27,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,22)(18,27)(19,29)(20,28)(21,26)(23,31)(24,32)(25,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=ghi,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, 9068299);