#9064214

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

4790 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=g,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, 8971410);