Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395181 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 226 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)(7,16)(8,20,22,29)(9,23)(10,14)(11,15)(12,18)(13,21,30,24)(17,26,33,36)(19,27)(25,28)(31,34,35,32);(1,2)(3,4)(5,7,10,25,11,19,9,18)(6,12,23,27,15,28,14,16)(8,17,21,32,22,33,24,34)(13,26,29,35,30,36,20,31);(2,4)(5,6)(7,12)(8,13)(9,14)(10,23)(11,15)(16,18)(17,26)(19,28)(20,24)(21,29)(22,30)(25,27)(31,34)(32,35)(33,36);(5,8,9,24,11,22,10,21)(6,13,14,20,15,30,23,29)(7,17,18,34,19,33,25,32)(12,26,16,31,28,36,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308197);