Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2510 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)(7,19)(8,14)(9,29)(10,18)(11,16)(12,17)(13,22)(15,30)(20,44)(21,48)(23,36)(24,37)(25,53)(26,55)(27,42)(28,40)(31,43)(32,39)(33,41)(34,51)(35,50)(38,57)(45,49)(46,65)(47,60)(52,61)(54,59)(56,63)(58,62)(64,67)(66,68);(5,7,10,22,12,23,11,24)(6,13,16,19,17,37,18,36)(8,25,27,51,28,20,32,47)(9,21,30,49,31,50,33,52)(14,34,39,53,40,60,42,44)(15,35,29,45,41,48,43,61)(26,54,56,67,58,46,57,66)(38,59,62,68,63,65,55,64);(2,4)(5,8)(6,14)(7,20)(9,26)(10,32)(11,27)(12,28)(13,34)(15,38)(16,42)(17,40)(18,39)(19,44)(21,46)(22,51)(23,25)(24,47)(29,55)(30,57)(31,58)(33,56)(35,59)(36,53)(37,60)(41,63)(43,62)(45,64)(48,65)(49,67)(50,54)(52,66)(61,68);(5,9,12,31)(6,15,17,41)(7,21,23,50)(8,26,28,58)(10,30,11,33)(13,35,37,48)(14,38,40,63)(16,29,18,43)(19,45,36,61)(20,46,25,54)(22,49,24,52)(27,56,32,57)(34,59,60,65)(39,62,42,55)(44,64,53,68)(47,66,51,67)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301575);