#10088978

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

4280 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)(11,16)(12,13)(14,21)(15,17)(18,20)(19,23)(22,24)(25,26,31,29)(27,35,37,32)(28,38,39,33)(30,34,36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,18)(14,22)(15,19)(17,23)(21,24)(25,27,31,37)(26,32,29,35)(28,36,39,30)(33,40,38,34);(1,3)(2,5)(4,8)(6,7)(10,17)(11,19)(12,22)(18,21)(25,26,31,29)(27,32,37,35)(28,33,39,38)(30,34,36,40);(1,2)(3,5)(4,7)(6,8)(9,12,15,22)(10,13,17,24)(11,20,19,14)(16,18,23,21)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,40)(34,35);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,21)(14,16)(15,24)(17,22)(18,19)(20,23)(25,29,31,26)(27,35,37,32)(28,33,39,38)(30,34,36,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,20)(10,18,17,21)(11,22,19,12)(13,23,24,16)(25,30,31,36)(26,34,29,40)(27,28,37,39)(32,33,35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9996174);