Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | #3449 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,20,15,21,35,48,16,22,36,49,13,19,33,46)(11,23,26,51,25,40,55,47,27,45,34,61,32,17,37,39)(12,18,30,43,31,44,60,69,28,41,56,65,29,42,58,67)(24,50,54,70,53,63,71,68,52,66,59,72,57,38,62,64);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,32)(14,34)(15,25)(16,27)(18,38)(19,45)(20,47)(21,23)(22,40)(26,36)(28,52)(29,57)(30,59)(31,53)(33,55)(35,37)(39,49)(41,63)(42,66)(43,68)(44,50)(46,51)(48,61)(54,56)(58,71)(60,62)(64,65)(67,70)(69,72);(9,12,15,31,16,28,13,29)(10,18,21,44,22,41,19,42)(11,24,25,53,27,52,32,57)(14,30,35,60,36,56,33,58)(17,38,23,50,40,63,45,66)(20,43,48,69,49,65,46,67)(26,54,55,71,34,59,37,62)(39,64,51,70,47,68,61,72);(12,28)(18,41)(24,52)(29,31)(30,56)(38,63)(42,44)(43,65)(50,66)(53,57)(54,59)(58,60)(62,71)(64,68)(67,69)(70,72)Polycyclic presentation:
2^9|a^2=f,c^2=h,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,e^b=ei,f^b=fgi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387327);