Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13867 |
| prime radical | #63090 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3679 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)(4,21,25,10)(5,26)(6,14)(7,12)(8,13)(9,18)(11,27)(16,35,43,48)(17,32)(19,33)(20,34)(22,31,47,57)(23,51,54,55)(24,37,30,40)(28,39)(29,38)(36,52,49,53)(41,44)(42,56,62,64)(45,59)(46,58)(50,63,61,60);(1,3)(2,9)(4,22)(5,17)(6,20)(7,18)(8,19)(10,35)(11,32)(12,33)(13,34)(14,15)(16,24)(21,48)(23,50)(25,47)(26,41)(27,46)(28,45)(29,44)(30,43)(31,37)(36,60)(38,59)(39,58)(40,57)(42,53)(49,63)(51,56)(52,62)(54,61)(55,64);(1,4)(2,10)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31)(11,36)(12,37)(13,21)(14,40)(15,35)(17,42)(18,22)(19,43)(20,47)(26,55)(27,52)(28,54)(29,53)(32,56)(33,48)(34,57)(38,49)(39,51)(41,60)(44,61)(45,62)(46,50)(58,63)(59,64);(1,5)(2,11)(3,17)(4,23)(6,29)(7,27)(8,28)(9,32)(10,36)(12,39)(13,38)(14,26)(15,41)(16,42)(18,46)(19,45)(20,44)(21,49)(22,50)(24,53)(25,54)(30,52)(31,56)(33,58)(34,59)(35,60)(37,51)(40,55)(43,62)(47,61)(48,63)(57,64)Polycyclic presentation:
2^9|a^2=h,e^2=i,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,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266882);