Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| 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 | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
596 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 596 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:
(1,2,3,4)(5,6,8,15,11,14,10,12)(7,17,20,29,21,30,19,13)(9,23,22,27,26,16,24,32)(18,33,34,31,35,28,25,36);(6,12)(7,18,21,35)(8,10)(9,24)(13,28,29,33)(14,15)(16,23)(17,31,30,36)(19,34,20,25)(22,26);(2,4)(5,7,8,19,11,21,10,20)(6,13,15,30,14,29,12,17)(9,25,22,35,26,34,24,18)(16,33,32,36,23,28,27,31);(6,14)(12,15)(13,29)(16,23)(17,30)(27,32)(28,33)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=eh,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250279, #72250290, #72250321, … |
| maximal normal subgroup of | #72250279, #72250290, #72250321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406619);