Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459338 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16338 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,18,9)(4,20,25,37)(5,26,28,11)(6,13,7,12)(10,24,38,23)(16,45,47,35)(17,44,49,33)(19,34,31,42)(21,43,50,32)(22,52,27,56)(29,40,30,41)(36,54,39,55)(46,61,64,62)(48,59,51,60)(53,58,57,63);(1,3,6,31,8,18,7,19)(2,9,12,42,14,15,13,34)(4,21,23,16,25,50,24,47)(5,17,30,48,28,49,29,51)(10,35,20,32,38,45,37,43)(11,33,41,59,26,44,40,60)(22,53,55,46,27,57,54,64)(36,61,52,58,39,62,56,63);(1,4)(2,10)(3,16)(5,27)(6,24)(7,23)(8,25)(9,32)(11,39)(12,37)(13,20)(14,38)(15,43)(17,46)(18,47)(19,50)(21,31)(22,28)(26,36)(29,54)(30,55)(33,58)(34,45)(35,42)(40,56)(41,52)(44,63)(48,53)(49,64)(51,57)(59,61)(60,62);(1,5,6,30,8,28,7,29)(2,11,12,41,14,26,13,40)(3,17,31,48,18,49,19,51)(4,22,23,55,25,27,24,54)(9,33,42,59,15,44,34,60)(10,36,20,52,38,39,37,56)(16,46,50,57,47,64,21,53)(32,58,45,62,43,63,35,61)Polycyclic presentation:
2^9|a^2=i,b^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366535);