#9061588

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

5135 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=g,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, 8968784);