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×#3487 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3504 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 3504 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)(7,12)(8,13)(9,14)(10,16)(11,15)(17,25)(18,26)(19,28)(20,29)(21,27)(22,30)(23,31)(24,32)(33,35)(34,36);(5,7,11,10)(6,12,15,16)(8,17,20,21)(9,18,23,24)(13,27,29,25)(14,26,31,32)(19,33,22,34)(28,36,30,35);(2,4)(5,8,10,21,11,20,7,17)(6,13,16,27,15,29,12,25)(9,22,18,34,23,19,24,33)(14,30,26,36,31,28,32,35);(5,9,11,23)(6,14,15,31)(7,18,10,24)(8,19,20,22)(12,26,16,32)(13,28,29,30)(17,33,21,34)(25,35,27,36)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=i,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 | #72306379, #72306519 |
| maximal normal subgroup of | #72306379, #72306519 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289635);