Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1326 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)(8,13)(9,21)(10,12)(11,17)(14,22)(16,28)(18,27)(19,32)(20,25)(23,31)(24,29)(26,33)(30,35)(34,36);(5,7)(6,11)(8,16)(9,17,22,31)(10,18)(12,23)(13,24)(14,15,21,27)(19,29,33,36)(20,30)(25,34)(26,28,32,35);(1,3,4,2)(6,12)(11,23)(13,25)(14,21)(15,27)(24,34)(26,32)(28,35);(5,8,10,20)(6,13,12,25)(7,16,18,30)(9,19,22,33)(11,24,23,34)(14,26,21,32)(15,28,27,35)(17,29,31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=i,b^a=be,c^a=ci,e^a=ei,g^a=ghi,e^b=eh,h^b=hi,g^e=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286452, #72286457, #72286495, … |
| maximal normal subgroup of | #72286452, #72286457, #72286495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400202);