Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343554 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62831 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,9,12,31)(4,19)(5,22)(6,27)(10,29,35,56)(11,38,40,60)(13,32,14,28)(15,43)(16,45)(20,47)(21,50)(23,52)(25,48)(26,53)(30,41,59,42)(33,36,57,37)(34,61,62,58)(39,64,63,55)(44,51)(49,54);(1,2)(3,10)(4,9)(5,11)(6,28)(7,13)(8,14)(12,24)(15,33)(16,34)(17,36)(18,37)(19,32)(20,29)(21,30)(22,42)(23,39)(25,38)(26,41)(27,31)(35,46)(40,53)(43,56)(44,61)(45,63)(47,57)(48,60)(49,55)(50,59)(51,58)(52,62)(54,64);(1,3,7,17)(2,10,13,36)(4,20,19,47)(5,23,26,16)(6,15,27,43)(8,18,24,46)(9,29,32,57)(11,39,41,34)(12,35,14,37)(21,51,48,49)(22,52,53,45)(25,54,50,44)(28,33,31,56)(30,58,60,55)(38,64,59,61)(40,63,42,62);(1,4,8,6)(2,9,14,28)(3,15,18,20)(5,21,22,25)(7,19,24,27)(10,33,37,29)(11,30,42,38)(12,31,13,32)(16,44,45,49)(17,43,46,47)(23,54,52,51)(26,48,53,50)(34,61,63,55)(35,57,36,56)(39,64,62,58)(40,59,41,60);(1,5,24,53)(2,11,12,40)(3,16,46,52)(4,21,27,50)(6,25,19,48)(7,26,8,22)(9,30,31,59)(10,34,35,62)(13,41,14,42)(15,44,47,51)(17,23,18,45)(20,49,43,54)(28,38,32,60)(29,55,56,64)(33,61,57,58)(36,39,37,63)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=hi,f^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250931);