Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,16,12,17,11,18)(7,19,22,39,23,38,24,13)(8,25,28,43,30,14,29,42)(9,31,33,46,34,15,35,45)(20,40,48,36,49,53,26,55)(21,50,32,37,51,44,52,61)(27,41,57,63,58,54,59,64)(47,65,66,68,67,62,56,60);(1,2)(3,4)(5,7,12,23)(6,13,17,39)(8,26,30,48)(9,32,34,52)(10,24,11,22)(14,40,25,53)(15,44,31,50)(16,38,18,19)(20,28,49,29)(21,35,51,33)(27,47,58,67)(36,42,55,43)(37,46,61,45)(41,60,54,68)(56,59,66,57)(62,64,65,63);(2,4)(5,8,12,30)(6,14,17,25)(7,20,23,49)(9,27,34,58)(10,29,11,28)(13,36,39,55)(15,41,31,54)(16,43,18,42)(19,40,38,53)(21,47,51,67)(22,26,24,48)(32,56,52,66)(33,59,35,57)(37,60,61,68)(44,62,50,65)(45,64,46,63);(5,9)(6,15,17,31)(7,21,23,51)(8,27)(10,33)(11,35)(12,34)(13,37)(14,41,25,54)(16,45,18,46)(19,44)(20,47,49,67)(22,32,24,52)(26,56,48,66)(28,57)(29,59)(30,58)(36,60)(38,50)(39,61)(40,62)(42,63,43,64)(53,65)(55,68)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334629);