#8458070

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

4413 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365266);