Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62928 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,22)(4,9,17,26)(5,10,6,19)(8,20,23,14)(13,27,30,32)(16,24,18,25)(21,28,31,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,55)(44,51)(45,50)(46,53)(47,52)(48,54)(49,57)(56,63)(58,61)(59,62)(60,64);(1,3,7,15)(2,8,11,23)(4,13,17,30)(5,14,6,20)(9,21,26,31)(10,22,19,12)(16,28,18,29)(24,27,25,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,55)(53,60)(54,61)(59,64)(62,63);(33,36,38,47)(34,40,41,52)(35,43,45,57)(37,46,48,59)(39,49,50,55)(42,53,54,62)(44,56,58,64)(51,60,61,63);(1,4)(2,9)(3,13)(5,18)(6,16)(7,17)(8,21)(10,25)(11,26)(12,27)(14,29)(15,30)(19,24)(20,28)(22,32)(23,31)(34,41)(39,50)(40,52)(42,54)(49,55)(51,61)(53,62)(60,63);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,26,25)(13,28,30,29)(21,27,31,32)(33,37,38,48)(34,42,41,54)(35,44,45,58)(36,46,47,59)(39,51,50,61)(40,53,52,62)(43,56,57,64)(49,60,55,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260005);