Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2093 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,14,11,15,10,26)(7,16,20,32,18,33,17,31)(8,21,24,30,22,36,25,13)(12,23,29,35,28,34,27,19);(1,2,3,4)(7,17,18,20)(8,22)(12,27,28,29)(13,30)(16,32,33,31)(19,23,35,34)(21,36)(24,25);(5,7)(6,12)(8,23)(9,20)(10,17)(11,18)(13,31)(14,29)(15,28)(16,21)(19,25)(22,34)(24,35)(26,27)(30,32)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250036, #72250041, #72250079, … |
| maximal normal subgroup of | #72250036, #72250041, #72250079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55252);