Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,13,10,14,23,15)(7,16,18,32,20,33,19,28)(8,21,24,12,25,29,34,30)(11,22,26,36,27,35,31,17);(5,7)(6,11,14,27)(8,22)(9,18)(10,20)(12,28,30,32)(13,26,15,31)(16,21,33,29)(17,34)(19,23)(24,36)(25,35);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240731, #72240733, #72240736, … |
| maximal normal subgroup of | #72240731, #72240733, #72240736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53712);