Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| 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 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,17,32)(8,19)(9,23)(10,13)(11,16,26,34)(12,22)(14,25)(18,29,24,33)(20,31)(21,30,36,27)(28,35);(1,2,3,4)(8,20)(11,26)(12,28)(16,34)(19,31)(21,36)(22,35)(27,30);(5,7)(6,11)(8,21)(9,24)(10,17)(12,29)(13,26)(14,30)(15,31)(16,35)(18,25)(19,32)(20,36)(22,34)(23,27)(28,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=ehi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214486, C2×#148043, #72215378, … |
| maximal normal subgroup of | #72214486, C2×#148043, #72215378, … |
| maximal quotient of | #72214240, #72214246, C2×#148043 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55239);