Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129437 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1193 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 142 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)(7,17,20,31)(8,21,24,29)(9,15,10,14)(11,16)(12,19)(13,23)(18,26)(22,30)(25,27)(28,34)(32,35,36,33);(1,2)(3,4)(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,28)(14,31)(15,17)(16,27)(21,32)(22,35)(23,33)(24,34)(26,30)(29,36);(2,4)(5,8,11,24)(6,13,16,30)(7,18,20,34)(9,23,10,22)(12,26,27,28)(14,21,15,29)(17,32,31,36)(19,35,25,33)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=ghi,d^c=di,e^c=ei,g^c=ghi,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54938);