Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13578 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 694 conjugacy classes of subgroups — too many to draw. Only its 130 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)(7,16,19,33)(8,21,23,32)(9,14,10,26)(11,15)(12,18,28,35)(13,25)(17,30,20,34)(22,29,36,27)(24,31);(1,2,3,4)(5,7,9,20,11,17,10,19)(6,12,14,29,15,27,26,28)(8,22,24,36,23,18,25,35)(13,30,21,33,31,16,32,34);(7,17)(12,27)(16,30)(18,22)(19,20)(28,29)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=h,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gh,h^a=hi,c^b=ci,d^b=di,e^b=eghi,e^d=ehi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284583, #72284600, #72284654, … |
| maximal normal subgroup of | #72284583, #72284600, #72284654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59063);