#10329316

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

4943 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,e^2=gh,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, 10236512);