Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2002 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)(15,31)(20,44)(21,35)(23,36)(24,37)(25,52)(26,54)(27,43)(28,40)(30,42)(32,41)(33,39)(34,51)(38,55)(45,50)(46,59)(47,60)(48,57)(49,61)(53,68)(56,63)(58,62)(64,65)(66,67);(5,7,10,22,12,23,11,24)(6,13,16,19,17,37,18,36)(8,25,27,51,28,20,33,47)(9,21,30,48,32,49,31,50)(14,34,39,52,40,60,43,44)(15,35,41,45,42,61,29,57)(26,53,55,66,56,46,58,65)(38,59,54,67,62,68,63,64);(2,4)(5,8)(6,14)(7,20)(9,26)(10,33)(11,27)(12,28)(13,34)(15,38)(16,43)(17,40)(18,39)(19,44)(21,46)(22,51)(23,25)(24,47)(29,54)(30,58)(31,55)(32,56)(35,59)(36,52)(37,60)(41,63)(42,62)(45,64)(48,66)(49,53)(50,65)(57,67)(61,68);(5,9,10,30,12,32,11,31)(6,15,16,41,17,42,18,29)(7,21,22,48,23,49,24,50)(8,26,33,58,28,56,27,55)(13,35,19,45,37,61,36,57)(14,38,43,63,40,62,39,54)(20,46,51,66,25,53,47,65)(34,59,44,64,60,68,52,67)Polycyclic presentation:
2^9|b^2=e,d^2=hi,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319045);