Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419794 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,10,16,12,18,11,17)(7,19,22,36,23,45,33,13)(8,24,26,14,28,39,27,40)(9,15,30,42,32,44,31,43)(20,34,48,61,49,37,51,46)(21,41,29,57,50,66,59,35)(25,52,54,38,56,64,55,65)(47,60,53,68,67,62,58,63);(5,7,11,22,12,23,10,33)(6,13,17,19,18,36,16,45)(8,20,27,48,28,49,26,51)(9,29,31,50,32,59,30,21)(14,37,24,46,40,34,39,61)(15,41,43,57,44,66,42,35)(25,53,55,67,56,58,54,47)(38,63,52,60,65,68,64,62);(5,8)(6,14)(7,20,23,49)(9,25)(10,27)(11,26)(12,28)(13,34,36,37)(15,38)(16,24)(17,39)(18,40)(19,46,45,61)(21,47,50,67)(22,51,33,48)(29,58,59,53)(30,55)(31,54)(32,56)(35,60,57,62)(41,63,66,68)(42,52)(43,64)(44,65);(2,4)(5,9,12,32)(6,15,18,44)(7,21)(8,25,28,56)(10,30,11,31)(13,35)(14,38,40,65)(16,42,17,43)(19,41)(20,47)(22,29)(23,50)(24,52,39,64)(26,54,27,55)(33,59)(34,60)(36,57)(37,62)(45,66)(46,63)(48,53)(49,67)(51,58)(61,68)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ef,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339201);