Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,21)(10,13)(11,17)(14,22)(16,27)(18,28)(19,26)(20,33)(23,30)(24,31)(25,32)(29,35)(34,36);(5,7)(6,11)(8,16)(9,17,22,31)(10,18)(12,23)(13,24)(14,15,21,28)(19,29)(20,30,32,36)(25,27,33,35)(26,34);(1,3,4,2)(5,8,10,19)(6,12,13,26)(7,16,18,29)(9,20,22,32)(11,23,24,34)(14,25,21,33)(15,27,28,35)(17,30,31,36);(6,13)(8,19)(11,24)(14,21)(15,28)(16,29)(20,32)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,e^2=i,b^a=be,d^a=di,e^a=ei,g^a=ghi,e^b=eh,h^b=hi,d^c=di,g^e=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317818, #72317821, #72317830, … |
| maximal normal subgroup of | #72317818, #72317821, #72317830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400232);