Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360368 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2344 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13904 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,18,39,19,9,29,33)(4,20,23,36,25,37,24,10)(5,26,27,40,28,11,30,38)(16,34,21,41,45,57,44,31)(17,32,47,59,48,42,46,58)(22,51,53,55,54,49,52,35)(43,62,63,61,64,60,50,56);(1,3,6,18,8,19,7,29)(2,9,13,33,12,15,14,39)(4,21,24,45,25,44,23,16)(5,17,27,46,28,48,30,47)(10,34,37,41,36,57,20,31)(11,32,38,58,26,42,40,59)(22,50,53,64,54,63,52,43)(35,60,51,61,55,62,49,56);(1,4)(2,10,12,36)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31,15,41)(11,35,26,55)(13,20,14,37)(17,43)(18,44)(19,45)(21,29)(27,52)(28,54)(30,53)(32,56,42,61)(33,57,39,34)(38,49,40,51)(46,63)(47,50)(48,64)(58,62,59,60);(1,5)(2,11)(3,17)(4,22)(6,27)(7,30)(8,28)(9,32)(10,35)(12,26)(13,38)(14,40)(15,42)(16,43)(18,46)(19,48)(20,49)(21,50)(23,52)(24,53)(25,54)(29,47)(31,56)(33,58)(34,60)(36,55)(37,51)(39,59)(41,61)(44,63)(45,64)(57,62)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269061);