Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14407 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,17,9,18,34,29,33)(4,10,21,37,22,38,30,36)(5,23,27,11,26,40,25,41)(15,42,45,31,19,47,44,58)(16,32,24,54,46,53,55,39)(20,49,51,35,52,61,56,60)(43,57,50,59,48,64,63,62);(1,3)(2,9)(4,19)(5,24)(6,29)(7,18)(8,17)(10,31)(11,39)(12,34)(13,33)(14,28)(15,22)(16,25)(20,50)(21,44)(23,53)(26,55)(27,46)(30,45)(32,40)(35,59)(36,42)(37,47)(38,58)(41,54)(43,56)(48,51)(49,57)(52,63)(60,62)(61,64);(1,4,8,21,7,22,6,30)(2,10,28,36,13,38,12,37)(3,15,29,44,18,19,17,45)(5,20,27,51,26,52,25,56)(9,31,34,47,33,58,14,42)(11,35,23,49,41,60,40,61)(16,43,55,63,46,48,24,50)(32,57,54,59,53,64,39,62);(1,5,8,27,7,26,6,25)(2,11,28,23,13,41,12,40)(3,16,29,55,18,46,17,24)(4,20,21,51,22,52,30,56)(9,32,34,54,33,53,14,39)(10,35,36,49,38,60,37,61)(15,43,44,63,19,48,45,50)(31,57,47,59,58,64,42,62)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292724);