#8766566

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

4922 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8673762);