#8662885

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

4615 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,f^2=h,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, 8570081);