#8718446

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

5137 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)(3,8)(4,5)(6,7)(9,10)(11,14)(12,16)(13,15)(17,18,23,27)(19,24,28,31)(20,25,29,30)(21,22,26,32);(1,3,6,5)(2,4,7,8)(9,11,15,16)(10,14,13,12)(17,19,23,28)(18,24,27,31)(20,21,29,26)(22,30,32,25);(1,3,6,5)(2,4,7,8)(9,12,15,14)(10,11,13,16)(19,28)(20,29)(24,31)(25,30);(2,7)(3,5)(11,16)(12,14)(17,20,23,29)(18,25,27,30)(19,21,28,26)(22,31,32,24);(1,4,6,8)(2,3,7,5)(9,13,15,10)(11,12,16,14)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,31)(27,32);(1,5,6,3)(2,4,7,8)(9,14,15,12)(10,11,13,16)(17,22)(18,26)(19,30)(20,24)(21,27)(23,32)(25,28)(29,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625642);