Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C24 | C4×C8 |
| Frattini, cosocle | C22×#131 | C22 |
| socle | C22 | #1282 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#131 ▷ C22×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,20,34,44,15,19,33,43,35,45,60,67)(11,21,23,46,25,47,51,68,24,48,31,57,52,69,58,71)(12,26,28,42,30,53,55,66,29,17,32,40,56,41,59,65)(16,36,37,49,39,61,62,27,38,50,63,54,64,70,72,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,19,38)(12,27,29,22)(13,31,33,23)(14,25)(15,24)(17,21)(18,37)(20,39,45,64)(26,48)(28,36)(30,54,56,49)(32,50)(34,58,60,51)(35,52)(40,46,42,57)(41,47)(43,63)(44,62)(53,69)(55,61)(59,70)(65,68,66,71)(67,72)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=h,h^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fi,g^a=gi,c^b=cg,d^b=df,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 682);