Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13975 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1393 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)(3,15,18,9,20,33,19,34)(4,21,25,37,26,40,24,10)(5,27,28,11,30,38,29,39)(16,31,43,55,44,56,22,35)(17,32,46,57,47,58,45,41)(23,36,50,48,52,60,51,53)(42,61,62,64,63,59,49,54);(1,3,7,18,8,20,6,19)(2,9,12,33,14,34,13,15)(4,22,25,16,26,43,24,44)(5,17,28,46,30,47,29,45)(10,35,21,31,37,55,40,56)(11,32,38,57,39,58,27,41)(23,49,50,42,52,62,51,63)(36,59,48,54,60,61,53,64);(1,4,6,24,8,26,7,25)(2,10,12,21,14,37,13,40)(3,16,19,22,20,44,18,43)(5,23,29,51,30,52,28,50)(9,31,33,55,34,56,15,35)(11,36,38,48,39,60,27,53)(17,42,45,49,47,63,46,62)(32,54,57,61,58,64,41,59);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,38)(13,27)(14,39)(15,41)(16,42)(18,46)(19,45)(20,47)(21,48)(22,49)(24,51)(25,50)(26,52)(31,54)(33,57)(34,58)(35,59)(37,60)(40,53)(43,62)(44,63)(55,61)(56,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272653);