Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,16,11,18,10,17)(7,19,24,38,23,37,22,13)(8,14,29,41,28,43,27,42)(9,30,34,15,33,45,32,46)(20,39,25,35,50,62,49,63)(21,51,53,47,52,36,31,44)(26,55,59,67,58,66,57,40)(48,65,60,68,56,61,54,64);(1,2)(3,4)(5,7,10,22,11,23,12,24)(6,13,16,19,18,38,17,37)(8,25,27,20,28,49,29,50)(9,31,32,52,33,53,34,21)(14,39,41,35,43,62,42,63)(15,44,45,51,46,47,30,36)(26,56,57,60,58,48,59,54)(40,65,55,68,67,61,66,64);(2,4)(5,8)(6,14)(7,20)(9,26)(10,29)(11,28)(12,27)(13,35)(15,40)(16,42)(17,41)(18,43)(19,39)(21,48)(22,25)(23,50)(24,49)(30,55)(31,60)(32,59)(33,58)(34,57)(36,61)(37,62)(38,63)(44,68)(45,66)(46,67)(47,64)(51,65)(52,56)(53,54);(5,9,11,33)(6,15,18,46)(7,21,23,52)(8,26,28,58)(10,34,12,32)(13,36,38,51)(14,40,43,67)(16,30,17,45)(19,47,37,44)(20,48,50,56)(22,53,24,31)(25,54,49,60)(27,59,29,57)(35,61,63,65)(39,64,62,68)(41,66,42,55)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326377);