Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454110 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
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 | |
2120 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)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,28)(15,34)(20,36)(21,51)(23,39)(24,38)(26,40)(27,59)(29,43)(30,42)(32,44)(33,46)(35,45)(37,55)(41,61)(47,54)(48,68)(49,65)(50,56)(52,67)(53,62)(57,60)(58,64)(63,66);(5,7,12,24)(6,13,18,39)(8,26,30,49)(9,21,35,55)(10,23,11,22)(14,40,43,65)(15,37,46,51)(16,38,17,19)(20,29,50,28)(25,56,42,36)(27,58,61,52)(31,62,45,47)(32,63,60,48)(33,54,34,53)(41,66,59,68)(44,64,57,67);(5,8,12,30)(6,14,18,43)(7,20,24,50)(9,32,35,60)(10,29,11,28)(13,36,39,56)(15,44,46,57)(16,42,17,25)(19,40,38,65)(21,52,55,58)(22,49,23,26)(27,34,61,33)(31,59,45,41)(37,66,51,68)(47,67,62,64)(48,54,63,53);(2,4)(5,9)(6,15)(7,21)(8,27)(10,34)(11,33)(12,35)(13,37)(14,41)(16,31)(17,45)(18,46)(19,47)(20,48)(22,54)(23,53)(24,55)(25,57)(26,58)(28,60)(29,32)(30,61)(36,64)(38,62)(39,51)(40,66)(42,44)(43,59)(49,52)(50,63)(56,67)(65,68)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361309);