Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501052 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,32,30,46,29,45,11,17,10,16,9,15,35,61)(7,13,33,60,21,39,52,65,22,40,20,38,19,37,54,67)(8,23,55,43,25,57,31,14,28,58,26,42,27,47,34,44)(18,48,62,63,24,56,53,36,51,68,49,41,50,66,59,64);(2,4)(5,7)(6,13)(8,24)(9,19)(10,20)(11,22)(12,33)(14,41)(15,37)(16,38)(17,40)(18,27)(21,30)(23,56)(25,51)(26,59)(28,50)(29,52)(31,49)(32,60)(34,62)(35,54)(36,43)(39,46)(42,64)(44,63)(45,65)(47,48)(53,55)(57,68)(58,66)(61,67);(5,8)(6,14)(7,18)(9,25)(10,31)(11,28)(12,34)(13,36)(15,42)(16,47)(17,44)(19,24)(20,53)(21,50)(22,51)(23,45)(26,29)(27,30)(32,57)(33,59)(35,55)(37,41)(38,66)(39,63)(40,64)(43,46)(48,65)(49,52)(54,62)(56,60)(58,61)(67,68);(2,4)(5,9,11,30)(6,15,17,46)(7,19,22,21)(8,25,28,27)(10,29,12,35)(13,37,40,39)(14,42,44,43)(16,45,32,61)(18,24,51,50)(20,52,33,54)(23,57,58,47)(26,34,55,31)(36,41,64,63)(38,65,60,67)(48,56,68,66)(49,59,62,53)Polycyclic presentation:
2^9|a^2=fi,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408253);