Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2084 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(6,12,14,16)(9,22)(13,29,31,27)(15,23,32,28)(17,33,34,35)(19,36)(21,25)(24,30);(1,2,4,3)(5,6,10,12,11,14,8,16)(7,17,18,35,20,34,26,33)(9,23,25,15,22,28,21,32)(13,30,29,19,31,24,27,36);(1,3,4,2)(5,7,8,18,11,20,10,26)(6,13,12,27,14,31,16,29)(9,24,21,36,22,30,25,19)(15,33,28,34,32,35,23,17)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250495, #72250498, #72250510, … |
| maximal normal subgroup of | #72250495, #72250498, #72250510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56158);