Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153155 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #30496 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(3,12)(4,17)(5,18)(7,23)(8,10)(9,27)(13,28)(14,16)(19,26)(21,22)(24,25)(30,32);(1,2,6,11)(3,13,16,28)(4,10,17,8)(5,19,22,9)(7,20,25,15)(12,29,14,31)(18,26,21,27)(23,30,24,32);(1,3)(2,7)(4,14)(5,20)(6,16)(8,24)(9,28)(10,23)(11,25)(12,17)(13,19)(15,22)(18,30)(21,32)(26,31)(27,29)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72283415, #72283820 |
| maximal normal subgroup of | #72283415, #72283820 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60352);