#8680000

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

4560 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587196);