Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2424 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,17,34)(4,19,23,10)(5,11,26,42)(6,27,7,12)(9,28,33,18)(15,43,45,35)(16,39,47,57)(20,31,46,56)(21,49,52,36)(22,37,29,38)(24,54,48,32)(25,41,30,40)(44,61,62,58)(50,55,63,64)(51,59,53,60);(1,3,6,18,8,17,7,28)(2,9,12,14,13,33,27,34)(4,20,22,15,23,46,29,45)(5,24,25,16,26,48,30,47)(10,35,37,56,19,43,38,31)(11,39,40,54,42,57,41,32)(21,44,51,63,52,62,53,50)(36,55,59,58,49,64,60,61);(1,4,6,22,8,23,7,29)(2,10,12,37,13,19,27,38)(3,15,18,46,17,45,28,20)(5,21,25,51,26,52,30,53)(9,31,14,35,33,56,34,43)(11,36,40,59,42,49,41,60)(16,44,48,63,47,62,24,50)(32,55,39,58,54,64,57,61);(1,5)(2,11)(3,16)(4,21)(6,25)(7,30)(8,26)(9,32)(10,36)(12,40)(13,42)(14,39)(15,44)(17,47)(18,48)(19,49)(20,50)(22,51)(23,52)(24,28)(27,41)(29,53)(31,55)(33,54)(34,57)(35,58)(37,59)(38,60)(43,61)(45,62)(46,63)(56,64);(3,17)(9,33)(14,34)(15,45)(16,47)(18,28)(20,46)(24,48)(31,56)(32,54)(35,43)(39,57)(44,62)(50,63)(55,64)(58,61)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346100);