Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
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 | |
1592 conjugacy classes of subgroups; only the 290 normal subgroups are shown.
This group has 1592 conjugacy classes of subgroups — too many to draw. Only its 290 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 48 points:
(2,4)(6,8)(17,18,22,25)(19,27,31,41)(20,24,32,39)(21,34,28,26)(23,35,37,30)(29,40,43,48)(33,45,42,38)(36,46,47,44);(9,10)(11,14)(12,13)(15,16)(17,19,28,35,22,31,21,30)(18,23,34,41,25,37,26,27)(20,29,42,46,32,43,33,44)(24,36,45,48,39,47,38,40);(1,2,3,4)(10,13)(14,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(5,6,7,8)(9,11,12,15)(10,14,13,16)(18,25)(20,32)(23,37)(26,34)(27,41)(29,43)(33,42)(44,46)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=f,d^2=ghi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 321619);