#8905381

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

4903 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812577);