Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1466 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)(3,12)(4,9)(5,18)(6,11)(7,14)(8,16)(10,20)(13,28)(15,26)(17,24)(19,23)(21,32)(22,30)(25,31)(27,29)(33,34,37,42,38,41,48,54)(35,39,45,51,44,50,57,61)(36,40,46,53,47,52,58,62)(43,49,56,60,55,59,63,64);(1,3,6,15)(2,7,11,23)(4,13,16,29)(5,19,20,14)(8,21,9,22)(10,26,18,12)(17,32,31,30)(24,28,25,27)(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);(1,4,6,16)(2,8,11,9)(3,13,15,29)(5,17,20,31)(7,21,23,22)(10,24,18,25)(12,27,26,28)(14,30,19,32)(33,36,37,46,38,47,48,58)(34,40,42,53,41,52,54,62)(35,43,45,56,44,55,57,63)(39,49,51,60,50,59,61,64);(1,4)(2,9)(3,13)(5,17)(6,16)(7,22)(8,11)(10,25)(12,28)(14,30)(15,29)(18,24)(19,32)(20,31)(21,23)(26,27)(35,44)(39,50)(43,55)(45,57)(49,59)(51,61)(56,63)(60,64);(4,16)(8,9)(13,29)(17,31)(21,22)(24,25)(27,28)(30,32)(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,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=dh,e^c=eh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252472);