Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13988 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1139 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)(3,9,17,33)(4,20,23,36)(5,11,26,39)(6,13)(7,14)(8,12)(10,24,37,25)(15,41)(16,45)(18,31,19,32)(21,29)(22,52)(27,38,28,40)(30,47)(34,44)(35,55)(42,61,62,59)(43,49)(46,58)(48,57)(50,54)(51,56,63,64)(53,60);(1,3)(2,9,12,33)(4,21)(5,16)(6,19)(7,18)(8,17)(10,34,37,41)(11,30,39,45)(13,32,14,31)(15,25)(20,49,36,29)(22,51)(23,43)(24,44)(26,47)(27,48)(28,46)(35,59,60,61)(38,58,40,57)(42,53)(50,64,52,56)(54,63)(55,62);(1,4)(2,10)(3,15)(5,22)(6,25)(7,24)(8,23)(9,29)(11,35)(12,37)(13,36)(14,20)(16,42)(17,44)(18,21)(19,43)(26,54)(27,55)(28,53)(30,56)(31,34)(32,41)(33,49)(38,50)(39,60)(40,52)(45,64)(46,63)(47,62)(48,51)(57,61)(58,59);(1,5,6,28,8,26,7,27)(2,11,14,38,12,39,13,40)(3,16,19,46,17,47,18,48)(4,22,25,53,23,54,24,55)(9,30,31,58,33,45,32,57)(10,35,20,50,37,60,36,52)(15,42,43,63,44,62,21,51)(29,56,34,59,49,64,41,61)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 273408);