#8107483

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

4506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(4,8)(6,7)(9,10,15,20)(11,16,22,23)(12,19,14,17)(13,18,24,21)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(1,2,5,3)(4,6,8,7)(9,11,15,22)(10,16,20,23)(12,18,14,21)(13,19,24,17)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(2,3)(6,7)(9,12,15,14)(10,17,20,19)(11,21,22,18)(13,16,24,23)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(1,3,5,2)(4,7,8,6)(9,13,15,24)(10,18,20,21)(11,17,22,19)(12,23,14,16)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,14,15,12)(10,19,20,17)(11,21,22,18)(13,16,24,23)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014679);