#8402459

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

4481 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:

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

Polycyclic presentation:

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

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, 8309655);