#8549660

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

5931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8456856);