Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453952 |
| 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 | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2136 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 2136 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)(8,12)(9,13)(10,14)(15,24)(16,22)(17,23)(18,31)(19,26)(20,27)(21,29)(25,32)(28,33)(30,34)(35,36);(1,2,4,3)(5,7,10,17)(6,11,14,23)(8,15,19,29)(9,16,20,30)(12,24,26,21)(13,22,27,34)(18,28,32,36)(25,35,31,33);(2,3)(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,25)(14,26)(16,28)(17,29)(20,32)(22,33)(23,24)(27,31)(30,36)(34,35);(1,3,4,2)(5,9)(6,13)(7,16)(8,18)(10,20)(11,22)(12,25)(14,27)(15,28)(17,30)(19,32)(21,33)(23,34)(24,35)(26,31)(29,36)Polycyclic presentation:
2^9|b^2=ei,d^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,d^c=de,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284265, #72284430, #72284802, … |
| maximal normal subgroup of | #72284265, #72284430, #72284802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361564);