Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,3,4)(6,11)(7,15)(8,18)(9,21)(12,25)(13,28)(14,24)(16,32)(17,31)(19,30)(22,26)(23,29)(27,36)(34,35);(5,6,8,11,10,14,18,24)(7,12,16,22,17,27,19,34)(9,13,20,23,21,29,33,28)(15,26,30,25,31,35,32,36);(5,7,9,15,10,17,21,31)(6,12,13,26,14,27,29,35)(8,19,20,32,18,16,33,30)(11,22,23,25,24,34,28,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312323, #72312327, #72312333, … |
| maximal normal subgroup of | #72312323, #72312327, #72312333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56472);