#9909761

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

4469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9816957);