#9712870

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

4547 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620066);