#8680003

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

4390 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,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, 8587199);