Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| 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 | |
1131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16)(7,17,22,44)(8,13,25,38)(9,26,29,14)(10,30,18,15)(12,32,35,21)(19,42,48,65)(20,39,49,43)(23,36,53,56)(24,54,45,37)(27,34,50,60)(28,40,31,41)(33,57,59,47)(46,64,66,61)(51,68,67,58)(52,63,55,62);(1,2)(3,4)(5,7,18,32,11,22,10,21)(6,12,30,44,16,35,15,17)(8,19,45,57,25,48,24,47)(9,27,31,20,29,50,28,49)(13,33,54,65,38,59,37,42)(14,39,41,34,26,43,40,60)(23,51,55,46,53,67,52,66)(36,61,63,58,56,64,62,68);(2,4)(5,8)(6,13)(7,19)(9,23)(10,24)(11,25)(12,33)(14,36)(15,37)(16,38)(17,42)(18,45)(20,46)(21,47)(22,48)(26,56)(27,51)(28,52)(29,53)(30,54)(31,55)(32,57)(34,58)(35,59)(39,61)(40,62)(41,63)(43,64)(44,65)(49,66)(50,67)(60,68);(5,9,10,31,11,29,18,28)(6,14,15,41,16,26,30,40)(7,20,21,50,22,49,32,27)(8,23,24,55,25,53,45,52)(12,34,17,43,35,60,44,39)(13,36,37,63,38,56,54,62)(19,46,47,67,48,66,57,51)(33,58,42,64,59,68,65,61)Polycyclic presentation:
2^9|a^2=i,b^2=ei,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,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, 335270);