Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #3203 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,9)(3,12,15,23)(4,10,16,24)(5,11,6,19)(8,20,21,14)(13,27,28,32)(17,25,18,26)(22,29,31,30)(33,34,37,42,38,41,48,54)(35,39,45,51,44,50,57,61)(36,40,47,53,46,52,58,62)(43,49,55,60,56,59,63,64);(1,3,7,15)(2,8,9,21)(4,13,16,28)(5,14,6,20)(10,22,24,31)(11,23,19,12)(17,29,18,30)(25,27,26,32)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,50)(42,51)(46,56)(47,55)(48,57)(52,59)(53,60)(54,61)(58,63)(62,64);(2,9)(4,16)(8,21)(11,19)(12,23)(13,28)(17,18)(29,30)(33,36,37,47,38,46,48,58)(34,40,42,53,41,52,54,62)(35,43,45,55,44,56,57,63)(39,49,51,60,50,59,61,64);(1,4)(2,10)(3,13)(5,18)(6,17)(7,16)(8,22)(9,24)(11,26)(12,27)(14,30)(15,28)(19,25)(20,29)(21,31)(23,32)(35,44)(36,46)(39,50)(40,52)(45,57)(47,58)(51,61)(53,62);(1,5,7,6)(2,11,9,19)(3,14,15,20)(4,17,16,18)(8,23,21,12)(10,25,24,26)(13,29,28,30)(22,27,31,32)(34,41)(39,50)(40,52)(42,54)(49,59)(51,61)(53,62)(60,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ei,f^a=fh,d^b=di,f^b=fh,d^c=dhi,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252800);