Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356478 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C42×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13810 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×H(3,ℤ/4) ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14,19,40)(4,20)(5,11)(6,12)(7,28)(8,13)(9,17,32,18)(10,22)(15,30,43,56)(16,36,45,60)(21,48)(23,34)(24,52,44,31)(25,37)(26,53)(27,38)(29,35)(33,50)(39,46,47,42)(41,55,61,64)(49,57,62,63)(51,58)(54,59);(1,3)(2,9,13,32)(4,15)(5,24,27,44)(6,17)(7,18)(8,19)(10,30,35,56)(11,36)(12,14,28,40)(16,25,45,26)(20,47,34,39)(21,49,51,62)(22,42)(23,43)(29,46)(31,53)(33,57)(37,52)(38,60)(41,50,61,54)(48,55)(58,64)(59,63);(1,4,6,22,8,23,7,29)(2,10,12,34,13,35,28,20)(3,15,17,42,19,43,18,46)(5,21,25,50,27,51,26,54)(9,30,14,39,32,56,40,47)(11,33,37,58,38,59,53,48)(16,41,24,49,45,61,44,62)(31,55,36,57,52,64,60,63);(1,5,6,25,8,27,7,26)(2,11,12,37,13,38,28,53)(3,16,18,44,19,45,17,24)(4,21,22,50,23,51,29,54)(9,31,40,60,32,52,14,36)(10,33,34,58,35,59,20,48)(15,41,46,62,43,61,42,49)(30,55,47,63,56,64,39,57)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,g^b=gi,f^c=fi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263752);