Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476175 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1074 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,16)(13,27)(14,19)(15,21)(18,30)(20,36)(22,37)(23,40)(24,31)(25,38)(26,35)(28,34)(29,32)(33,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,21,14,22,30,29)(10,15,18,17,19,32,27,37)(12,23,25,24,26,20,28,39)(16,31,34,40,35,33,38,36);(2,6)(5,8)(9,12)(10,16)(11,20)(13,28)(14,26)(15,31)(17,36)(18,38)(19,35)(21,24)(22,23)(25,30)(27,34)(29,39)(32,33)(37,40);(3,7)(5,8)(9,11)(10,17)(12,24)(13,29)(14,22)(15,18)(16,33)(19,37)(20,28)(21,30)(23,25)(26,39)(27,32)(31,35)(34,40)(36,38)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383563);