Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14016 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,9,18,32,19,33,17,31)(4,20,24,36,23,40,25,10)(5,11,26,37,28,38,27,39)(15,34,43,47,42,58,21,29)(16,30,44,55,46,56,45,57)(22,50,53,35,51,48,52,60)(41,61,63,54,62,59,49,64);(1,3,7,18,8,19,6,17)(2,9,13,32,14,33,12,31)(4,21,24,15,23,43,25,42)(5,16,26,44,28,46,27,45)(10,34,20,47,36,58,40,29)(11,30,37,55,38,56,39,57)(22,49,53,41,51,63,52,62)(35,59,48,64,60,61,50,54);(1,4,6,24,8,23,7,25)(2,10,13,40,14,36,12,20)(3,15,17,43,19,42,18,21)(5,22,27,53,28,51,26,52)(9,29,32,58,33,47,31,34)(11,35,37,50,38,60,39,48)(16,41,45,63,46,62,44,49)(30,54,55,61,56,64,57,59);(1,5)(2,11)(3,16)(4,22)(6,27)(7,26)(8,28)(9,30)(10,35)(12,39)(13,37)(14,38)(15,41)(17,45)(18,44)(19,46)(20,48)(21,49)(23,51)(24,53)(25,52)(29,54)(31,57)(32,55)(33,56)(34,59)(36,60)(40,50)(42,62)(43,63)(47,64)(58,61)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275419);