Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1282 |
| prime radical | C24 | #97 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,5,3,4,7,6,8)(9,10,13,18,15,20,31,42,14,19,30,41,32,43,57,66)(11,21,23,44,24,46,50,67,25,45,29,55,51,68,56,71)(12,17,26,37,28,39,53,63,27,38,52,62,54,64,70,72)(16,33,34,58,35,59,60,22,36,47,40,49,61,69,65,48);(1,3,6,7,4,8,5,2)(9,11)(10,16)(12,22,27,48)(13,29)(14,25)(15,24)(17,21,38,45)(18,40)(19,36)(20,35)(23,30)(26,33,52,47)(28,49,54,58)(31,56)(32,51)(34,41)(37,55,62,44)(39,46,64,68)(42,65)(43,61)(50,57)(53,59,70,69)(60,66)(63,71,72,67)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=h,e^2=f,h^2=i,b^a=bc,c^a=cf,e^a=egi,c^b=cg,d^b=dfi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 695);