Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 683 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,18,23,34)(8,24)(9,15)(10,16)(11,27)(12,17)(13,21,29,22)(14,25)(19,28,36,30)(20,33,35,32)(26,31);(5,7,11,22,12,23,10,21)(6,13,16,18,17,29,27,34)(8,19,25,35,26,36,9,20)(14,30,24,32,15,28,31,33);(5,8,12,26)(6,14,17,15)(7,19,23,36)(9,10,25,11)(13,28,29,30)(16,31,27,24)(18,32,34,33)(20,21,35,22);(1,3,4,2)(5,9,10,26,12,25,11,8)(6,15,16,31,17,14,27,24)(7,20,21,36,23,35,22,19)(13,28,18,33,29,30,34,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=fgh,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289540);