#9160872

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

4908 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=i,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, 9068068);