Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13971 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1839 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)(3,15,20,9)(4,21,24,37)(5,27,29,11)(6,13)(7,14)(8,12)(10,25,38,26)(16,35)(17,32)(18,34,19,33)(22,41)(23,49)(28,40,30,39)(31,45)(36,52)(42,47)(43,59,62,64)(44,56)(46,57)(48,58)(50,61,63,55)(51,60)(53,54);(1,3,6,19,8,20,7,18)(2,9,13,34,12,15,14,33)(4,22,26,16,24,45,25,44)(5,17,30,46,29,47,28,48)(10,35,37,31,38,56,21,41)(11,32,40,57,27,42,39,58)(23,50,52,43,51,63,53,62)(36,59,60,55,54,64,49,61);(1,4)(2,10)(3,16)(5,23)(6,26)(7,25)(8,24)(9,31)(11,36)(12,38)(13,37)(14,21)(15,41)(17,43)(18,22)(19,45)(20,44)(27,54)(28,53)(29,51)(30,52)(32,55)(33,35)(34,56)(39,49)(40,60)(42,61)(46,63)(47,62)(48,50)(57,64)(58,59);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,32)(10,36)(12,27)(13,40)(14,39)(15,42)(16,43)(18,48)(19,46)(20,47)(21,49)(22,50)(24,51)(25,53)(26,52)(31,55)(33,58)(34,57)(35,59)(37,60)(38,54)(41,61)(44,62)(45,63)(56,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272330);