#8703422

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

5224 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^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, 8610618);