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 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,15,12,17,9,27)(7,18,20,29,21,16,10,13)(8,22,23,34,26,30,25,35)(14,28,33,24,32,36,31,19);(1,2)(3,4)(5,7,9,10,12,21,11,20)(6,13,15,18,17,29,27,16)(8,19,23,28,26,24,25,36)(14,30,31,34,32,22,33,35);(2,4)(5,8,9,23,12,26,11,25)(6,14,15,31,17,32,27,33)(7,19,10,28,21,24,20,36)(13,22,18,35,29,30,16,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250133, #72250135, #72250158, … |
| maximal normal subgroup of | #72250133, #72250135, #72250158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55919);