#8406724

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

4546 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313920);