Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500112 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
572 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 154 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)(5,6,9,24,10,14,25,15)(7,11,18,36,16,27,33,29)(8,19,22,13,23,12,21,32)(17,30,35,28,20,26,34,31);(1,2,4,3)(5,7)(6,11)(8,20)(9,18)(10,16)(12,30)(13,31)(14,27)(15,29)(17,23)(19,26)(21,35)(22,34)(24,36)(25,33)(28,32);(6,12)(8,21)(9,25)(11,26)(13,24)(14,19)(15,32)(17,34)(18,33)(20,35)(22,23)(27,30)(28,36)(29,31);(1,3,4,2)(7,16)(11,27)(17,20)(18,33)(26,30)(28,31)(29,36)(34,35)Polycyclic presentation:
2^9|a^2=gh,b^2=e,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237246, #72239800, #72239803, … |
| maximal normal subgroup of | #72237246, #72239800, #72239803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412877);