Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622850 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#447 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #3573 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,15)(3,16,20,43)(4,11,21,38)(5,25,29,42)(6,14,7,13)(9,31,35,18)(10,30,37,19)(12,27,32,28)(17,45,46,62)(22,50,49,61)(23,41,24,40)(26,44,55,34)(33,56,58,47)(36,54,60,48)(39,52,57,53)(51,63,64,59);(1,3)(2,9)(4,17)(5,26)(6,19)(7,30)(8,20)(10,32)(11,33)(12,37)(13,34)(14,44)(15,35)(16,25)(18,28)(21,46)(22,51)(23,48)(24,54)(27,31)(29,55)(36,57)(38,58)(39,60)(40,59)(41,63)(42,43)(45,50)(47,53)(49,64)(52,56)(61,62);(1,3)(2,10)(4,17)(5,18)(6,30)(7,19)(8,20)(9,32)(11,36)(12,35)(13,43)(14,16)(15,37)(21,46)(22,47)(23,54)(24,48)(25,44)(26,28)(27,55)(29,31)(33,57)(34,42)(38,60)(39,58)(40,62)(41,45)(49,56)(50,63)(51,53)(52,64)(59,61);(1,4)(2,11)(3,17)(5,22)(6,23)(7,24)(8,21)(9,33)(10,36)(12,39)(13,40)(14,41)(15,38)(16,45)(18,47)(19,48)(20,46)(25,50)(26,51)(27,52)(28,53)(29,49)(30,54)(31,56)(32,57)(34,59)(35,58)(37,60)(42,61)(43,62)(44,63)(55,64);(4,21)(11,38)(17,46)(22,49)(23,24)(33,58)(36,60)(39,57)(40,41)(45,62)(47,56)(48,54)(50,61)(51,64)(52,53)(59,63)Polycyclic presentation:
2^9|a^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,h^a=hi,f^b=fh,h^b=hi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530047);