Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153119 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13348 |
| prime radical | #33546 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 524 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:
(1,2)(3,12,16,24)(4,8)(5,9)(6,11)(7,14,23,17)(10,20)(13,28,15,29)(18,25)(19,27)(21,30,22,31)(26,32);(7,21)(12,28)(13,15)(22,23)(24,29)(30,31);(1,3)(2,7)(4,17)(5,13)(6,16)(8,24)(9,21)(10,22)(11,23)(12,27)(14,19)(15,20)(18,31)(25,29)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=fh,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279311, #72339871 |
| maximal normal subgroup of | #72279311, #72339871 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60379);