#8969347

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

5106 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^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, 8876543);