#8492944

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

4437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8400140);