#8991889

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

4906 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)(3,7)(4,5)(6,8)(17,18,22,24)(19,25,27,21)(20,23,30,28)(26,31,32,29);(3,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,26)(22,27)(23,29)(24,25)(28,31)(30,32);(1,3,5,8)(2,6,4,7)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,21,27,25)(20,28,30,23)(26,31,32,29);(3,8)(6,7)(9,11,12,16)(10,13,14,15)(17,18)(19,21)(20,28)(22,24)(23,30)(25,27)(26,31)(29,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,30)(18,23,24,28)(19,26,27,32)(21,29,25,31);(1,4,5,2)(3,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8899085);