Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623063 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#481 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #60462 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,15,39)(4,9,18,32)(5,21,22,53)(6,26,24,11)(8,28,30,16)(10,35,36,25)(14,40,42,61)(17,37,27,41)(19,48,46,64)(20,51,47,34)(23,31,45,38)(29,54,55,43)(33,58,57,50)(44,59,52,62)(49,56,63,60);(5,22)(10,36)(13,39)(16,28)(19,46)(21,53)(23,45)(25,35)(33,57)(37,41)(40,61)(43,54)(48,64)(49,63)(50,58)(59,62);(1,3,6,27,7,15,24,17)(2,8,11,38,12,30,26,31)(4,14,20,52,18,42,47,44)(5,23,25,28,22,45,35,16)(9,29,34,60,32,55,51,56)(10,37,21,39,36,41,53,13)(19,49,50,54,46,63,58,43)(33,59,48,61,57,62,64,40);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,29)(10,33)(11,34)(12,32)(13,40)(15,42)(16,43)(17,44)(21,48)(22,46)(23,49)(24,47)(25,50)(26,51)(27,52)(28,54)(30,55)(31,56)(35,58)(36,57)(37,59)(38,60)(39,61)(41,62)(45,63)(53,64);(4,18)(9,32)(14,42)(19,46)(20,47)(29,55)(33,57)(34,51)(40,61)(43,54)(44,52)(48,64)(49,63)(50,58)(56,60)(59,62)Polycyclic presentation:
2^9|a^2=i,c^2=fh,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,g^b=gi,f^c=fi,g^c=ghi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530260);