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 | |
798 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 218 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)(5,6,9,12,11,14,8,15)(7,17,19,30,20,27,22,13)(10,24,23,32,26,16,21,28)(18,29,35,31,36,34,25,33);(6,12)(7,18)(8,9)(10,23)(13,29)(14,15)(16,24)(17,33)(19,25)(20,36)(21,26)(22,35)(27,31)(30,34);(2,4)(5,7,8,19,11,20,9,22)(6,13,15,17,14,30,12,27)(10,25,21,18,26,35,23,36)(16,29,32,31,24,34,28,33);(6,14)(12,15)(13,30)(16,24)(17,27)(28,32)(29,34)(31,33)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,f^2=i,g^2=hi,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=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250509, #72250521, #72250538, … |
| maximal normal subgroup of | #72250509, #72250521, #72250538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406558);