#9128640

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

4855 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,f^2=gh,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, 9035836);