#9061205

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

5338 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^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, 8968401);