#10366269

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

4652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 10273465);