Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14064 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1233 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)(3,13)(4,18,21,9)(5,22)(6,25)(7,12)(8,16)(10,23)(11,27)(14,39,42,36)(15,43)(17,38)(19,49,51,57)(20,33,26,35)(24,52)(28,46,31,41)(29,45)(30,47)(32,50,48,53)(34,54)(37,44)(40,56,62,55)(58,61,60,63)(59,64);(1,3,6,16,7,17,27,47)(2,8,11,13,12,30,25,38)(4,14,26,46,21,42,20,41)(5,15,23,44,24,45,54,64)(9,31,35,39,18,28,33,36)(10,29,22,37,34,43,52,59)(19,40,53,63,51,62,50,61)(32,56,49,60,48,55,57,58);(1,4)(2,9)(3,14)(5,19)(6,26)(7,21)(8,28)(10,32)(11,35)(12,18)(13,36)(15,40)(16,46)(17,42)(20,27)(22,49)(23,53)(24,51)(25,33)(29,55)(30,31)(34,48)(37,58)(38,39)(41,47)(43,56)(44,63)(45,62)(50,54)(52,57)(59,60)(61,64);(1,5)(2,10,12,34)(3,15)(4,19)(6,23)(7,24)(8,29,30,43)(9,32,18,48)(11,22,25,52)(13,37,38,59)(14,40)(16,44)(17,45)(20,50)(21,51)(26,53)(27,54)(28,55,31,56)(33,57,35,49)(36,58,39,60)(41,61)(42,62)(46,63)(47,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^c=ei,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278213);