Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33669 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 433 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,21,26,22,28,38,27)(19,29,32,24,30,44,46,42)(20,33,36,25,37,23,34,40)(31,45,47,39,35,43,48,41);(3,9)(6,12)(10,15)(13,16)(18,23)(19,30)(20,34)(21,38)(24,41)(25,26)(27,40)(28,33)(29,45)(31,47)(35,48)(36,37)(39,42)(43,44);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,24,28,42)(20,35)(21,32)(22,30)(23,39,33,41)(25,45,40,43)(26,44,27,29)(31,37)(34,47)(36,48)(38,46);(2,7)(6,12)(8,14)(13,16)(19,30)(24,42)(29,44)(31,35)(32,46)(39,41)(43,45)(47,48)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286391, #72286482, #72286518, … |
| maximal normal subgroup of | #72286391, #72286482, #72286518, … |
| maximal quotient of | #72343328, #72343451, #72343475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391130);