Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7339802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #48088 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23873 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 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)(3,13,17,38)(4,10)(5,23,25,37)(6,12)(7,9)(8,16,28,26)(11,27,32,24)(14,39,41,34)(15,44)(18,29,43,56)(19,31)(20,33)(21,51,49,62)(22,36)(30,46)(35,54,58,53)(40,45)(42,64)(47,55)(48,61)(50,59)(52,60)(57,63);(1,3,7,17)(2,8,9,28)(4,18,19,43)(5,15,25,46)(6,26,20,16)(10,34,31,39)(11,30,32,44)(12,38,33,13)(14,22,41,50)(21,52,49,48)(23,55,37,40)(24,47,27,45)(29,36,56,59)(35,61,58,60)(42,53,63,54)(51,57,62,64);(2,9)(4,19)(6,20)(8,28)(11,32)(14,41)(16,26)(21,49)(24,27)(30,44)(34,39)(36,59)(42,63)(45,47)(51,62)(60,61);(1,4,20,50,7,19,6,22)(2,10,33,59,9,31,12,36)(3,14,16,43,17,41,26,18)(5,21,24,53,25,49,27,54)(8,29,13,39,28,56,38,34)(11,35,37,62,32,58,23,51)(15,42,47,52,46,63,45,48)(30,57,40,61,44,64,55,60);(1,5)(2,11)(3,15)(4,21)(6,27)(7,25)(8,30)(9,32)(10,35)(12,23)(13,40)(14,42)(16,47)(17,46)(18,48)(19,49)(20,24)(22,54)(26,45)(28,44)(29,57)(31,58)(33,37)(34,60)(36,51)(38,55)(39,61)(41,63)(43,52)(50,53)(56,64)(59,62)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7247000);