Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14518 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,34,20,42,18,9)(4,21,25,10,26,37,24,38)(5,27,29,40,31,11,30,41)(16,32,46,43,47,50,22,35)(17,39,28,33,48,44,49,59)(23,51,53,36,55,62,54,56)(45,58,57,63,64,61,52,60);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,42)(4,22,24,47,26,46,25,16)(5,28,29,48,31,49,30,17)(10,35,37,32,38,43,21,50)(11,39,40,59,27,44,41,33)(23,45,53,57,55,64,54,52)(36,61,51,63,56,58,62,60);(1,4,7,25,8,26,6,24)(2,10,12,37,14,38,13,21)(3,16,19,46,20,47,18,22)(5,23,29,53,31,55,30,54)(9,32,15,43,34,50,42,35)(11,36,41,62,27,56,40,51)(17,45,28,57,48,64,49,52)(33,58,44,63,59,61,39,60);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,36)(12,41)(13,40)(14,27)(15,44)(16,45)(18,49)(19,28)(20,48)(21,51)(22,52)(24,54)(25,53)(26,55)(32,58)(34,59)(35,60)(37,62)(38,56)(39,42)(43,63)(46,57)(47,64)(50,61)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295475);