#9065743

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

4989 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:

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

Polycyclic presentation:

2^9|a^2=gi,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^c=egi,e^d=egi,f^d=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, 8972939);