Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1390 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)(7,17)(8,23)(9,28)(10,32)(11,16)(12,20)(13,26)(14,30)(15,29)(18,37)(19,47)(21,36)(22,35)(24,38)(25,40)(27,39)(31,57)(33,46)(34,50)(41,58)(42,45)(43,48)(44,65)(49,60)(51,61)(52,53)(54,56)(55,63)(59,68)(62,66)(64,67);(5,7,11,22)(6,12)(8,18)(9,19,31,50)(10,20,29,21)(13,37,40,33)(14,34)(15,17)(16,36)(23,42,39,61)(24,44)(25,45)(26,51)(27,46)(28,43)(30,48,58,49)(32,35)(38,62,63,59)(41,47)(52,64,56,65)(53,66)(54,68)(55,67)(57,60);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,27)(12,33)(14,38)(15,39)(16,40)(17,42)(19,44)(20,45)(21,51)(22,46)(23,32)(26,29)(28,56)(30,53)(31,55)(34,59)(35,61)(36,37)(41,63)(43,64)(47,62)(48,66)(49,68)(50,67)(52,57)(54,58)(60,65);(5,9,29,58,11,31,10,30)(6,14,32,57,16,41,15,28)(7,19,21,49,22,50,20,48)(8,24,26,54,27,55,25,53)(12,34,35,60,36,47,17,43)(13,38,23,52,40,63,39,56)(18,44,51,68,46,67,45,66)(33,59,61,65,37,62,42,64)Polycyclic presentation:
2^9|a^2=h,b^2=efghi,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274295);