Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3548 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 3548 conjugacy classes of subgroups — too many to draw. Only its 354 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,14)(10,13)(15,21)(16,23)(17,22)(18,25)(19,26)(20,27)(24,28)(29,32)(30,33)(31,35)(34,36);(5,7,10,17)(6,11,13,22)(8,15,18,28)(9,16,20,30)(12,24,25,21)(14,23,27,33)(19,29,31,36)(26,34,35,32);(2,4)(5,8,9,19,10,18,20,31)(6,12,14,26,13,25,27,35)(7,15,16,29,17,28,30,36)(11,21,23,32,22,24,33,34);(8,18)(9,20)(12,25)(14,27)(15,28)(16,30)(21,24)(23,33)Polycyclic presentation:
2^9|b^2=i,c^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240678, #72251718, #72318305, … |
| maximal normal subgroup of | #72240678, #72251718, #72318305, … |
| maximal quotient of | #72238535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289633);