Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,16,11,18,10,17)(7,19,24,39,23,38,22,13)(8,14,29,42,28,44,27,43)(9,30,35,15,34,47,33,48)(20,40,25,36,52,62,51,63)(21,37,31,45,55,65,54,49)(26,56,58,41,57,46,32,59)(50,66,68,67,60,61,53,64);(1,2)(3,4)(5,7,10,22,11,23,12,24)(6,13,16,19,18,39,17,38)(8,25,27,20,28,51,29,52)(9,31,33,21,34,54,35,55)(14,40,42,36,44,62,43,63)(15,45,47,65,48,49,30,37)(26,50,32,53,57,60,58,68)(41,66,46,67,59,61,56,64);(2,4)(5,8,11,28)(6,14,18,44)(7,20,23,52)(9,32,34,58)(10,29,12,27)(13,36,39,63)(15,46,48,56)(16,43,17,42)(19,40,38,62)(21,53,55,68)(22,25,24,51)(26,35,57,33)(30,59,47,41)(31,60,54,50)(37,64,65,67)(45,61,49,66);(5,9,11,34)(6,15,18,48)(7,21,23,55)(8,26,28,57)(10,35,12,33)(13,37,39,65)(14,41,44,59)(16,30,17,47)(19,49,38,45)(20,50,52,60)(22,31,24,54)(25,53,51,68)(27,58,29,32)(36,61,63,66)(40,64,62,67)(42,56,43,46)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383634);