Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,13)(8,23)(9,29)(10,18)(11,16)(12,17)(14,26)(15,30)(19,44)(20,47)(21,42)(22,37)(24,51)(25,55)(27,39)(28,40)(31,43)(32,36)(33,41)(34,46)(35,49)(38,57)(45,59)(48,62)(50,61)(52,56)(53,68)(54,60)(58,63)(64,66)(65,67);(1,2)(3,4)(5,7)(6,13)(8,24)(9,20)(10,32)(11,21)(12,22)(14,34)(15,35)(16,42)(17,37)(18,36)(19,27)(23,51)(25,53)(26,46)(28,54)(29,47)(30,49)(31,50)(33,48)(38,59)(39,44)(40,60)(41,62)(43,61)(45,57)(52,67)(55,68)(56,65)(58,66)(63,64);(5,8,10,26,12,27,11,28)(6,14,16,23,17,40,18,39)(7,19,21,46,22,24,32,54)(9,25,30,56,31,57,33,58)(13,34,36,44,37,60,42,51)(15,38,29,52,41,55,43,63)(20,45,48,65,50,53,49,66)(35,59,61,64,62,68,47,67);(1,3,4,2)(5,9,12,31)(6,15,17,41)(7,20,22,50)(8,25,27,57)(10,30,11,33)(13,35,37,62)(14,38,40,55)(16,29,18,43)(19,45,24,53)(21,48,32,49)(23,52,39,63)(26,56,28,58)(34,59,60,68)(36,61,42,47)(44,64,51,67)(46,65,54,66)Polycyclic presentation:
2^9|c^2=f,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330722);