#10171927

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

4462 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)(3,8,4,6)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26,31,33)(27,34,30,32)(28,37,38,29)(35,40,39,36);(3,4)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,31,30)(26,32,33,34)(28,35,38,39)(29,40,37,36);(1,3,5,4)(2,6,7,8)(10,14)(15,17)(16,20)(22,23)(26,33)(28,38)(32,34)(35,39);(3,4)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,38)(26,29,33,37)(27,35,30,39)(32,40,34,36);(1,4,5,3)(2,6,7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,29)(26,28)(27,36)(30,40)(31,37)(32,39)(33,38)(34,35);(1,2)(3,6)(4,8)(5,7)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30,31,27)(26,34,33,32)(28,39,38,35)(29,36,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=hi,e^2=i,f^2=g,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, 10079123);