#8699298

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

4438 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8606494);