Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
742 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)(7,16,18,32)(8,21)(9,13)(10,15)(11,17,26,34)(12,24)(14,25)(19,33,20,28)(22,31,35,27)(23,30)(29,36);(5,7,9,19,10,18,25,20)(6,11,13,27,15,26,14,31)(8,22,24,34,23,35,36,17)(12,28,30,16,29,33,21,32);(1,2,3,4)(8,23)(12,29)(16,32)(17,34)(21,30)(22,35)(24,36)(28,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,e^b=ei,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226840 |
| maximal normal subgroup of | #72226840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54652);