Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147593 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10965 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,16,18,8)(5,19)(6,10)(7,13,22,29)(9,20)(12,27)(14,26)(15,25)(17,24,32,31)(21,30,23,28);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,21)(10,22)(15,20)(16,26)(17,28)(18,29)(19,23)(24,25)(27,31)(30,32);(3,12)(5,20)(7,21)(9,19)(11,25)(13,28)(14,15)(17,32)(22,23)(24,31)(26,27)(29,30)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gh,f^c=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72240494, #72258266, #72267963, … |
| maximal normal subgroup of | #72240494, #72258266, #72267963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54789);