#8985283

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

4554 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)(17,18,23,26)(19,27,30,21)(20,25,22,24)(28,31,29,32);(3,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,21,26,27)(20,28,22,29)(24,31,25,32);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(17,19)(18,21)(20,28)(22,29)(23,30)(24,31)(25,32)(26,27);(3,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,23,22)(18,24,26,25)(19,28,30,29)(21,31,27,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,19)(20,31)(22,32)(23,27)(24,28)(25,29)(26,30);(1,4,5,2)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,22,23,20)(18,25,26,24)(19,29,30,28)(21,32,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=ghi,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, 8892479);