Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 842 conjugacy classes of subgroups — too many to draw. Only its 234 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:
(5,6,12,29,25,34,8,15,11,16,24,14,10,17,9,26)(7,18,23,30,28,36,20,31,19,32,27,13,22,33,21,35);(1,2)(3,4)(5,7)(6,13,16,30)(8,23)(9,27)(10,28)(11,19)(12,20)(14,32,29,18)(15,33,26,36)(17,35,34,31)(21,24)(22,25);(2,4)(6,14)(8,24)(9,12)(10,25)(13,18)(15,34)(16,29)(17,26)(20,27)(21,23)(22,28)(30,32)(31,36)(33,35);(7,19)(13,30)(18,32)(20,21)(22,28)(23,27)(31,35)(33,36)Polycyclic presentation:
2^9|a^2=eh,b^2=efh,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224928, #72224931, #72226822, … |
| maximal normal subgroup of | #72224928, #72224931, #72226822, … |
| maximal quotient of | #72224918 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406741);