Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2380 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13832 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1063 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)(3,8,15,31)(4,17,20,47)(5,22)(6,12)(7,11)(9,21,34,39)(10,24)(13,38)(14,42,23,54)(16,30,44,58)(18,48)(19,33,51,62)(25,36)(26,35)(27,37)(28,50)(29,43,57,45)(32,41)(40,63)(46,52,61,53)(49,59)(55,56)(60,64);(1,3)(2,8,11,31)(4,18)(5,23,25,14)(6,16)(7,15)(9,32,34,38)(10,29)(12,30,37,58)(13,39)(17,28,47,48)(19,49,51,64)(20,50)(21,41)(22,54)(24,45,26,43)(27,44)(33,60)(35,57)(36,42)(40,53,55,52)(46,63)(56,61)(59,62);(1,4)(2,9,11,34)(3,13)(5,19)(6,21)(7,20)(8,28,31,48)(10,33,35,62)(12,17,37,47)(14,40)(15,41)(16,18)(22,46,36,61)(23,55)(24,52)(25,51)(26,53)(27,39)(29,56,57,63)(30,38,58,32)(42,59,54,60)(43,64)(44,50)(45,49);(1,5)(2,10)(3,14)(4,19)(6,26)(7,25)(8,29)(9,33)(11,35)(12,22)(13,40)(15,23)(16,45)(17,46)(18,49)(20,51)(21,53)(24,27)(28,56)(30,54)(31,57)(32,59)(34,62)(36,37)(38,60)(39,52)(41,55)(42,58)(43,44)(47,61)(48,63)(50,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,h^a=hi,c^b=cef,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,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, 264571);