Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups; only the 302 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 302 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(10,12)(13,14,19,24)(15,25,29,35)(16,30,31,38)(17,23,18,22)(20,28,36,27)(21,32,37,33)(26,34,40,44)(39,41,43,42);(5,6)(7,8)(13,15,18,28,19,29,17,27)(14,20,23,35,24,36,22,25)(16,26,33,42,31,40,32,41)(21,34,30,43,37,44,38,39);(2,4)(6,8)(13,16,17,32,19,31,18,33)(14,21,22,38,24,37,23,30)(15,26,27,41,29,40,28,42)(20,34,25,39,36,44,35,43);(9,10,11,12)(16,31)(21,37)(26,40)(30,38)(32,33)(34,44)(39,43)(41,42)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=e,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319325);