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×Q8 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1063 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:
(5,6,11,18,12,17,8,24)(7,19,22,32,23,34,20,29)(9,15,25,14,10,16,26,33)(13,28,31,21,30,36,35,27);(1,2)(3,4)(5,7)(6,13)(8,20)(9,27)(10,21)(11,22)(12,23)(14,32)(15,34)(16,19)(17,30)(18,35)(24,31)(25,28)(26,36)(29,33);(2,4)(5,8,12,11)(6,14,17,33)(7,20,23,22)(9,26,10,25)(13,29,30,32)(15,24,16,18)(19,31,34,35)(21,28,27,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=i,g^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, 54940);