Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6444406 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,3,4)(5,6,11,17,12,18,10,16)(7,19,23,13,24,37,22,38)(8,25,30,14,28,42,29,41)(9,15,33,45,34,46,32,44)(20,35,26,39,50,54,51,47)(21,48,31,36,53,43,52,62)(27,55,60,40,58,66,59,65)(49,61,56,63,57,64,68,67);(2,4)(5,7)(6,13)(8,26)(9,31)(10,23)(11,22)(12,24)(14,39)(15,43)(16,37)(17,19)(18,38)(20,30)(21,33)(25,54)(27,57)(28,51)(29,50)(32,53)(34,52)(35,42)(36,45)(40,64)(41,47)(44,62)(46,48)(49,58)(55,67)(56,60)(59,68)(61,65)(63,66);(5,8,12,28)(6,14)(7,20)(9,27,34,58)(10,29,11,30)(13,35,38,54)(15,40)(16,25)(17,42)(18,41)(19,47,37,39)(21,49)(22,51)(23,26)(24,50)(31,56)(32,59,33,60)(36,61,62,64)(43,63,48,67)(44,55)(45,66)(46,65)(52,68)(53,57);(5,9,10,32,12,34,11,33)(6,15,16,44,18,46,17,45)(7,21,22,52,24,53,23,31)(8,27,29,59,28,58,30,60)(13,36,19,48,38,62,37,43)(14,40,25,55,41,65,42,66)(20,49,51,68,50,57,26,56)(35,61,47,67,54,64,39,63);(8,28)(14,41)(20,50)(25,42)(26,51)(27,58)(29,30)(35,54)(39,47)(40,65)(49,57)(55,66)(56,68)(59,60)(61,64)(63,67)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372877);