Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,21)(12,26,28,36)(13,17)(14,19)(15,20)(16,24)(18,29,33,30)(22,31)(23,38,27,40)(25,35)(32,39,34,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,27)(13,24)(14,22)(15,25)(17,21)(18,34)(19,35)(20,31)(23,28)(26,38)(29,39)(30,37)(32,33)(36,40);(1,4,5,2)(3,7,8,6)(10,17)(11,22)(14,15)(16,31)(18,33)(21,35)(23,37)(24,25)(27,39)(29,30)(32,38)(34,40);(3,8)(6,7)(9,12)(10,18)(11,23)(13,28)(14,29)(15,30)(16,32)(17,33)(19,26)(20,36)(21,34)(22,37)(24,27)(25,39)(31,38)(35,40)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326582);