#9169330

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

5044 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9076526);