#9065301

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

5023 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,f^2=gh,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, 8972497);