#9105338

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

5156 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)(11,15)(12,16)(13,14)(17,18,22,25)(19,23,28,31)(20,29,21,24)(26,32,27,30);(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,16,12)(17,19,22,28)(18,23,25,31)(20,27,21,26)(24,32,29,30);(2,3)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20,22,21)(18,24,25,29)(19,26,28,27)(23,30,31,32);(2,3)(4,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,20)(18,24,25,29)(19,27,28,26)(23,30,31,32);(1,3,5,2)(4,7,8,6)(9,10)(11,12)(13,14)(15,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,31)(29,30);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,32)(28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=gh,e^2=hi,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, 9012534);