#9063908

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

5530 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)(11,12,16,15)(17,18,23,21)(19,25,20,24)(22,29,30,26)(27,31,28,32);(1,2)(3,4)(5,6)(7,8)(9,10,13,14)(11,15,16,12)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32);(2,6)(3,7)(10,14)(11,16)(18,21)(24,25)(26,29)(31,32);(3,7)(4,8)(11,16)(12,15)(17,20,23,19)(18,25,21,24)(22,27,30,28)(26,31,29,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,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, 8971104);