Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1466 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,13,9,14,21,29,10,11,20,24,22,25,33,36)(7,15,17,31,18,23,34,28,16,30,19,35,32,12,26,27);(1,2,3,4)(6,11)(7,16)(14,25)(17,19)(18,32)(26,34)(27,28)(31,35);(5,7,9,18,10,16,22,32)(6,12,25,30,11,23,14,15)(8,19,21,26,20,17,33,34)(13,28,36,31,24,27,29,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,e^2=h,h^2=i,b^a=bd,c^a=ch,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239962, #72250587, #72250956, … |
| maximal normal subgroup of | #72239962, #72250587, #72250956, … |
| maximal quotient of | #72240004 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34194);