Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1067 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)(5,6,10,16,11,14,9,15)(7,17,20,30,21,13,18,31)(8,12,24,27,23,28,25,32)(19,34,35,33,22,26,36,29);(6,12)(7,18)(9,10)(13,29)(14,28)(15,32)(16,27)(17,33)(19,35)(20,21)(22,36)(24,25)(26,30)(31,34);(2,4)(5,7,11,21)(6,13,14,17)(8,22,23,19)(9,20,10,18)(12,26,28,34)(15,31,16,30)(24,35,25,36)(27,33,32,29)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312755, #72313178, #72337435, … |
| maximal normal subgroup of | #72312755, #72313178, #72337435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55052);