#9105360

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

5058 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)(11,13)(12,16)(14,15)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31);(1,2,5,3)(4,6,8,7)(10,15)(11,16)(17,19,22,25)(18,23,26,21)(20,27,28,29)(24,31,32,30);(2,3)(6,7)(9,11,14,16)(10,12,15,13)(20,28)(24,32)(27,29)(30,31);(2,3)(4,8)(9,12,14,13)(10,11,15,16)(17,19)(18,23)(20,29)(21,26)(22,25)(24,30)(27,28)(31,32);(1,3,5,2)(4,7,8,6)(9,10)(11,12)(13,16)(14,15)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,29)(26,32);(1,4)(2,6)(3,7)(5,8)(9,13,14,12)(10,16,15,11)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,29,32,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 9012556);