Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1387 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1387 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,9,14,18,29)(7,16,20,32,17,12,27,30)(8,13,22,31,23,11,24,26)(19,25,33,35,21,28,34,36);(1,2,3,4)(6,11,14,13)(7,17)(12,28,16,25)(15,31,29,26)(19,21)(20,27)(30,35,32,36)(33,34);(5,7,18,27,9,17,10,20)(6,12,29,32,14,16,15,30)(8,21,22,34,23,19,24,33)(11,25,26,35,13,28,31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fhi,d^c=dh,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312378, #72312382, #72312390, … |
| maximal normal subgroup of | #72312378, #72312382, #72312390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54687);