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 | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 435 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)(17,18,21,27,23,28,22,39)(19,29,32,41,30,45,33,25)(20,34,35,44,38,24,37,26)(31,43,46,48,36,42,47,40);(18,24)(20,35)(21,22)(25,40)(26,39)(27,44)(28,34)(29,42)(31,46)(32,33)(36,47)(37,38)(41,48)(43,45);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25,28,41)(20,36)(21,32)(22,33)(23,30)(24,40,34,48)(26,43,44,42)(27,29,39,45)(31,38)(35,47)(37,46);(2,7)(6,12)(8,14)(13,16)(19,30)(25,41)(29,45)(31,36)(32,33)(40,48)(42,43)(46,47)Polycyclic presentation:
2^9|a^2=f,c^2=fg,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 | #72321989, #72322020, #72322039, … |
| maximal normal subgroup of | #72321989, #72322020, #72322039, … |
| maximal quotient of | #72343313, #72343418, #72343551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391082);