Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14100 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 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)(3,8,15,30)(4,17,20,9)(5,10,22,36)(6,24)(7,11)(12,27)(13,39)(14,42)(16,37,25,31)(18,32)(19,33)(21,34,26,38)(23,52,51,35)(28,41)(29,43)(40,61,62,55)(44,56)(45,46)(47,49)(48,57,63,64)(50,60)(53,59)(54,58);(1,3)(2,8,11,30)(4,18)(5,14)(6,25)(7,15)(9,32,17,46)(10,29,36,42)(12,37,24,31)(13,21)(16,27)(19,48)(20,45)(22,43)(23,53)(26,41)(28,38,39,34)(33,57,47,64)(35,59,52,56)(40,50)(44,51)(49,63)(54,62)(55,60,61,58);(1,4)(2,9)(3,13)(5,19)(6,26)(7,20)(8,28)(10,33)(11,17)(12,38)(14,40)(15,41)(16,45)(18,25)(21,27)(22,49)(23,54)(24,34)(29,55)(30,39)(31,46)(32,37)(35,60)(36,47)(42,61)(43,62)(44,63)(48,53)(50,51)(52,58)(56,64)(57,59);(1,5,6,23,7,22,27,51)(2,10,12,35,11,36,24,52)(3,14,25,53,15,43,16,44)(4,19,26,54,20,49,21,50)(8,29,37,59,30,42,31,56)(9,33,38,60,17,47,34,58)(13,40,18,48,41,62,45,63)(28,55,32,57,39,61,46,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 279983);