#8702654

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

4618 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8609850);