#8726725

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

4973 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)(9,10,13,11)(12,15,16,14)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32);(2,5)(6,7)(10,11)(14,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(19,20)(24,25)(27,28)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,20,23,19)(18,25,21,24)(22,27,30,28)(26,31,29,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32);(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=gi,e^2=gi,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, 8633921);