Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,13,9,14,19,26,10,11,18,22,20,23,30,33)(7,12,16,24,17,25,29,34,15,21,27,31,28,32,35,36);(1,2,3,4)(6,11)(7,15)(14,23)(16,27)(17,28)(24,31)(29,35)(34,36);(5,7,9,17,10,15,20,28)(6,12,14,25,11,21,23,32)(8,16,19,29,18,27,30,35)(13,24,26,34,22,31,33,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,e^2=h,h^2=i,b^a=bd,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239959, #72239962, #72250264, … |
| maximal normal subgroup of | #72239959, #72239962, #72250264, … |
| maximal quotient of | #72240001, #72307732 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34154);