Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152774 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13090 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3498 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,8,14,4,19)(3,9,17,24,18,26,30,32)(5,12,22,10,7,11,20,27)(15,29,21,28,23,31,16,25);(1,3,6,23)(2,9,10,24)(4,15,8,18)(5,21,22,30)(7,16,20,17)(11,25,13,29)(12,28,19,31)(14,26,27,32);(1,4)(2,10)(3,15)(5,22)(6,8)(7,20)(9,24)(11,19)(12,13)(14,27)(16,17)(18,23)(21,30)(25,31)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72286416, #72287234, #72287238 |
| maximal normal subgroup of | #72286416, #72287234, #72287238 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59972);