Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563333 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(5,6)(7,21)(8,28)(9,35)(10,16)(11,19)(12,18)(13,20)(14,25)(15,32)(17,38)(22,40)(23,41)(24,57)(26,43)(27,42)(29,52)(30,59)(31,51)(33,48)(34,47)(36,62)(37,50)(39,46)(44,54)(45,61)(49,56)(53,67)(55,66)(58,68)(60,64)(63,65);(1,2)(3,4)(5,7,11,25,13,27,12,26)(6,14,18,21,20,43,19,42)(8,29,32,22,34,54,33,55)(9,36,10,24,38,56,37,23)(15,44,28,40,48,52,47,66)(16,49,35,57,50,62,17,41)(30,53,39,63,61,58,31,60)(45,64,46,68,59,65,51,67);(5,8,11,32,13,34,12,33)(6,15,18,28,20,48,19,47)(7,22,26,29,27,55,25,54)(9,30,10,39,38,61,37,31)(14,40,42,44,43,66,21,52)(16,45,35,46,50,59,17,51)(23,53,56,60,24,58,36,63)(41,64,62,67,57,65,49,68);(2,4)(5,9,12,37,13,38,11,10)(6,16,19,17,20,50,18,35)(7,23,25,36,27,24,26,56)(8,30,33,31,34,61,32,39)(14,41,21,49,43,57,42,62)(15,45,47,51,48,59,28,46)(22,53,54,63,55,58,29,60)(40,64,52,68,66,65,44,67);(5,10,11,38,13,37,12,9)(6,17,18,16,20,35,19,50)(7,24,25,56,27,23,26,36)(8,31,32,30,34,39,33,61)(14,41,21,49,43,57,42,62)(15,46,28,59,48,51,47,45)(22,53,54,63,55,58,29,60)(40,65,52,67,66,64,44,68)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=gi,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470531);