Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1116 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)(7,15)(8,12)(9,21,22,13)(10,14)(11,17,24,33)(16,30)(18,29)(19,27,34,25)(20,26)(23,31,35,36)(28,32);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,23)(13,15)(14,24)(19,31)(20,32)(21,29)(22,33)(25,30)(26,35)(27,28)(34,36);(5,8,10,20)(6,12,14,26)(7,16,18,32)(9,19,22,34)(11,23,24,35)(13,27,21,25)(15,28,29,30)(17,31,33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281750, #72303107, #72303162, … |
| maximal normal subgroup of | #72281750, #72303107, #72303162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53790);