#9063287

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

4857 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)(3,5)(4,7)(6,8)(10,12)(14,15)(17,18,23,21)(19,24,27,29)(20,25,30,31)(22,28,32,26);(1,2,6,8)(3,5,7,4)(9,10,13,12)(11,15,16,14)(17,19)(18,24)(20,26)(21,29)(22,25)(23,27)(28,30)(31,32);(1,3,6,7)(2,4,8,5)(9,11,13,16)(10,14,12,15)(18,21)(19,27)(20,30)(22,32);(1,4,6,5)(2,3,8,7)(9,12,13,10)(11,15,16,14)(17,20)(18,25)(19,28)(21,31)(22,29)(23,30)(24,32)(26,27);(1,2)(3,4)(5,7)(6,8)(17,21,23,18)(19,29,27,24)(20,25,30,31)(22,28,32,26);(1,5,6,4)(2,7,8,3)(17,22,23,32)(18,26,21,28)(19,25,27,31)(20,29,30,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8970483);