Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1936 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10821 |
| 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 | |
451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,9,15,8,36)(3,16,19,39,22,46,21,58)(4,23,27,48,30,42,29,44)(5,31,33,47,34,14,7,12)(10,20,38,32,40,56,50,18)(11,35,43,25,45,28,60,26)(17,41,53,63,55,64,24,37)(49,54,59,51,61,57,62,52);(1,3,7,20,8,21,34,18,9,22,33,56,6,19,5,32)(2,10,14,39,13,38,12,46,15,40,31,58,36,50,47,16)(4,24,28,52,29,55,25,57,30,53,35,51,27,17,26,54)(11,41,44,49,43,63,23,59,45,64,48,61,60,37,42,62);(1,4,5,25,6,27,33,28,9,30,34,26,8,29,7,35)(2,11,12,42,36,60,14,48,15,45,47,23,13,43,31,44)(3,17,18,51,19,53,20,57,22,55,32,52,21,24,56,54)(10,37,16,49,50,64,58,62,40,63,46,61,38,41,39,59)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314268 |
| maximal normal subgroup of | #72314268 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54136);