Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1972 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10863 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,16)(4,23,27,56)(5,13)(6,31)(7,33)(8,35)(9,15)(10,18)(11,26,42,59)(12,30)(14,29)(17,49,52,36)(19,47)(20,39)(21,37)(22,48)(24,40,57,64)(25,44,34,61)(28,41,58,43)(32,38)(45,53)(46,51,55,60)(50,62,54,63);(1,3,7,20,29,53,30,19,9,22,5,18,8,21,6,32)(2,10,12,16,35,47,33,37,15,39,31,48,14,38,13,45)(4,24,25,51,58,17,26,50,27,57,34,60,28,52,59,54)(11,40,43,62,61,36,23,55,42,64,41,63,44,49,56,46);(1,4,6,26,8,28,5,25,9,27,30,59,29,58,7,34)(2,11,13,23,14,44,31,43,15,42,33,56,35,61,12,41)(3,17,19,51,21,24,20,54,22,52,32,60,53,57,18,50)(10,36,37,62,38,40,16,46,39,49,45,63,47,64,48,55)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314212 |
| maximal normal subgroup of | #72314212 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54300);