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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3022 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)(5,6,11,34,12,17,10,16)(7,18,33,13,22,37,21,38)(8,23,28,14,26,49,27,41)(9,15,32,58,30,43,31,44)(19,35,51,61,46,60,24,39)(20,42,57,36,47,62,29,56)(25,48,54,40,55,66,53,65)(45,59,67,68,52,64,50,63);(2,4)(5,7)(6,13)(8,24)(9,29)(10,33)(11,21)(12,22)(14,39)(15,42)(16,37)(17,38)(18,34)(19,27)(20,31)(23,35)(25,52)(26,51)(28,46)(30,57)(32,47)(36,44)(40,64)(41,61)(43,62)(45,55)(48,63)(49,60)(50,53)(54,67)(56,58)(59,65)(66,68);(5,8,12,26)(6,14)(7,19)(9,25,30,55)(10,27,11,28)(13,35,38,60)(15,40)(16,23)(17,41)(18,39,37,61)(20,45)(21,24)(22,46)(29,50)(31,53,32,54)(33,51)(34,49)(36,59,56,64)(42,63,62,68)(43,65)(44,48)(47,52)(57,67)(58,66);(5,9,10,31,12,30,11,32)(6,15,16,44,17,43,34,58)(7,20,21,29,22,47,33,57)(8,25,27,53,26,55,28,54)(13,36,18,42,38,56,37,62)(14,40,23,48,41,65,49,66)(19,45,24,50,46,52,51,67)(35,59,39,63,60,64,61,68);(8,26)(9,30)(14,41)(15,43)(19,46)(20,47)(23,49)(24,51)(27,28)(29,57)(31,32)(35,60)(36,56)(39,61)(42,62)(44,58)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,d^2=h,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=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372883);