Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| 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 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,14,24)(4,8,15,22)(5,17,19,9)(7,18,21,13)(12,25,26,32)(16,29,28,23)(20,30,31,27)(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);(1,3,6,14)(2,7,10,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(9,24,17,11)(16,30,28,27)(23,32,29,25)(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);(1,4,6,15)(2,8,10,22)(3,12,14,26)(5,16,19,28)(7,20,21,31)(9,23,17,29)(11,25,24,32)(13,27,18,30)(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)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32)(35,44)(39,50)(43,55)(45,57)(49,59)(51,61)(56,63)(60,64);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(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=g,f^2=h,g^2=i,b^a=bf,e^a=ei,f^a=fh,d^b=di,f^b=fh,d^c=dh,e^c=eh,e^d=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252468);