Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,12,17)(7,19,23,37)(8,14,28,41)(9,30,33,42)(10,18,11,16)(13,24,36,22)(15,31,43,32)(20,46,25,44)(21,35,51,61)(26,54,56,62)(27,40,29,39)(34,48,60,49)(38,55,63,57)(45,50,58,52)(47,66,53,59)(64,68,65,67);(1,2,3,4)(5,7,10,22,12,23,11,24)(6,13,16,19,17,36,18,37)(8,25,27,49,28,20,29,48)(9,21,31,50,33,51,32,52)(14,34,39,44,41,60,40,46)(15,35,42,45,43,61,30,58)(26,53,55,68,56,47,57,67)(38,59,62,64,63,66,54,65);(5,8)(6,14)(7,20,23,25)(9,26)(10,27)(11,29)(12,28)(13,34,36,60)(15,38)(16,39)(17,41)(18,40)(19,44,37,46)(21,47,51,53)(22,48,24,49)(30,54)(31,55)(32,57)(33,56)(35,59,61,66)(42,62)(43,63)(45,64,58,65)(50,67,52,68);(5,9,10,31,12,33,11,32)(6,15,16,42,17,43,18,30)(7,21,22,50,23,51,24,52)(8,26,27,55,28,56,29,57)(13,35,19,45,36,61,37,58)(14,38,39,62,41,63,40,54)(20,47,48,67,25,53,49,68)(34,59,44,64,60,66,46,65)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,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,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320417);