Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14137 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1647 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)(3,15)(4,20,25,37)(5,26)(6,13)(7,12)(8,14)(9,18)(10,24,38,23)(11,27)(16,42,45,31)(17,46)(19,33)(21,50,44,35)(22,52,55,36)(28,40)(29,39)(30,34)(32,48)(41,47)(43,62,64,59)(49,54,60,53)(51,57,63,61)(56,58);(1,3,6,18,8,19,7,30)(2,9,12,15,14,34,13,33)(4,21,23,16,25,44,24,45)(5,17,27,47,29,48,28,56)(10,35,20,42,38,50,37,31)(11,32,26,41,40,46,39,58)(22,51,53,43,55,63,54,64)(36,59,49,61,52,62,60,57);(1,4,6,23,8,25,7,24)(2,10,13,37,14,38,12,20)(3,16,18,44,19,45,30,21)(5,22,27,53,29,55,28,54)(9,31,33,50,34,42,15,35)(11,36,39,60,40,52,26,49)(17,43,47,63,48,64,56,51)(32,57,58,62,46,61,41,59);(1,5,6,27,8,29,7,28)(2,11,12,26,14,40,13,39)(3,17,18,47,19,48,30,56)(4,22,23,53,25,55,24,54)(9,32,15,41,34,46,33,58)(10,36,20,49,38,52,37,60)(16,43,44,63,45,64,21,51)(31,57,35,59,42,61,50,62)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281646);