Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394358 |
| 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 | |
2528 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)(7,12)(8,13)(9,25)(10,28)(11,15)(14,26)(16,30)(17,31)(18,42)(19,34)(20,46)(21,48)(22,51)(23,36)(24,55)(27,39)(29,38)(32,44)(33,45)(35,53)(37,54)(40,63)(41,65)(43,60)(47,59)(49,68)(50,58)(52,62)(56,61)(57,64)(66,67);(5,7,11,19)(6,12,15,34)(8,21)(9,18,27,43)(10,20,30,45)(13,31)(14,32,38,59)(16,33,28,46)(17,23)(22,49)(24,50)(25,42,39,60)(26,44,29,47)(35,57)(36,48)(37,58)(40,52)(41,54)(51,63)(53,66)(55,65)(56,64)(61,67)(62,68);(2,4)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,31)(14,35)(15,36)(16,37)(18,40)(19,21)(20,41)(25,51)(26,53)(27,52)(28,55)(29,56)(30,54)(32,57)(33,58)(34,48)(38,61)(39,62)(42,63)(43,49)(44,64)(45,50)(46,65)(47,66)(59,67)(60,68);(5,9,11,27)(6,14,15,38)(7,18,19,43)(8,22,23,52)(10,29,30,26)(12,32,34,59)(13,35,36,61)(16,39,28,25)(17,40,21,49)(20,47,45,44)(24,56,54,53)(31,57,48,67)(33,60,46,42)(37,62,55,51)(41,66,50,64)(58,68,65,63)Polycyclic presentation:
2^9|b^2=ei,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 301557);