#8722285

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

4909 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,f^2=gi,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, 8629481);