Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453974 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2010 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 2010 conjugacy classes of subgroups — too many to draw. Only its 218 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:
(2,3)(5,6)(7,11)(8,18)(9,21)(10,14)(12,19)(13,24)(15,32)(16,30)(17,27)(20,31)(22,26)(23,29)(25,33)(28,35)(34,36);(1,2,4,3)(5,7,10,17)(6,11,14,27)(8,15,20,34)(9,22,24,16)(12,28,29,25)(13,30,21,26)(18,32,31,36)(19,35,23,33);(2,3)(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,31)(14,29)(16,35)(17,34)(18,21)(19,24)(22,33)(26,36)(27,28)(30,32);(1,3,4,2)(9,24)(13,21)(16,22)(18,31)(19,23)(26,30)(32,36)(33,35)Polycyclic presentation:
2^9|b^2=ei,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312508, #72312596 |
| maximal normal subgroup of | #72312508, #72312596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361553);