Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1307 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1307 conjugacy classes of subgroups — too many to draw. Only its 234 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)(5,6,11,17,12,18,10,16)(7,19,21,13,22,14,8,23)(9,25,28,32,29,15,27,33)(20,34,24,36,35,31,26,30);(5,7)(6,13)(8,11)(9,26)(10,21)(12,22)(14,16)(15,31)(17,19)(18,23)(20,27)(24,29)(25,34)(28,35)(30,32)(33,36);(1,3,4,2)(5,8,11,7,12,21,10,22)(6,14,17,23,18,19,16,13)(9,24,27,20,29,26,28,35)(15,31,32,36,25,34,33,30);(5,9)(6,15)(7,20)(8,24)(10,28)(11,27)(12,29)(13,30)(14,31)(16,33)(17,32)(18,25)(19,34)(21,26)(22,35)(23,36)Polycyclic presentation:
2^9|a^2=h,c^2=eghi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297607);