#8750828

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

4365 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658024);