#9910904

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

4552 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)(3,4,8,6)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,31,34)(27,35,29,32)(28,30,39,33)(36,40,38,37);(1,3,5,8)(2,6,7,4)(11,17)(14,21)(18,23)(22,24)(25,27,31,29)(26,32,34,35)(28,36,39,38)(30,37,33,40);(1,3)(2,4)(5,8)(6,7)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,31,39)(26,33,34,30)(27,36,29,38)(32,40,35,37);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,30)(27,36)(29,38)(31,39)(32,37)(33,34)(35,40);(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,29,31,27)(26,32,34,35)(28,38,39,36)(30,37,33,40);(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30)(26,28)(27,37)(29,40)(31,33)(32,36)(34,39)(35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818100);