Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483882 |
| 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 | |
455 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,6,9,13,10,12,15,16)(17,18,21,26,22,28,38,27)(19,29,32,24,30,43,45,41)(20,33,36,25,37,23,34,40)(31,44,47,39,35,42,46,48);(18,23)(20,34)(21,38)(24,39)(25,26)(27,40)(28,33)(29,42)(31,46)(32,45)(35,47)(36,37)(41,48)(43,44);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,24,28,41)(20,35)(21,32)(22,30)(23,39,33,48)(25,44,40,42)(26,43,27,29)(31,37)(34,47)(36,46)(38,45);(2,7)(6,12)(8,14)(13,16)(19,30)(24,41)(29,43)(31,35)(32,45)(39,48)(42,44)(46,47)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,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216810, #72216824, #72216898, … |
| maximal normal subgroup of | #72216810, #72216824, #72216898, … |
| maximal quotient of | C2×#483888, #72318633, #72318666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391084);