Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 | |
1718 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 1718 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:
(5,6,9,14,10,13,25,24)(7,11,18,27,17,15,23,30)(8,19,21,12,20,28,22,29)(16,32,33,26,34,36,35,31);(1,2)(3,4)(5,7,9,23,10,17,25,18)(6,11,24,27,13,15,14,30)(8,16,22,33,20,34,21,35)(12,26,28,32,29,31,19,36);(2,4)(5,6)(7,15)(8,12)(9,24)(10,13)(11,17)(14,25)(16,31)(18,27)(19,21)(20,29)(22,28)(23,30)(26,34)(32,35)(33,36);(5,8,9,22,10,20,25,21)(6,12,24,28,13,29,14,19)(7,16,23,33,17,34,18,35)(11,26,27,32,15,31,30,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,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, 308199);