Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13343 |
| commutator, Frattini, cosocle | #532 | C23 |
| 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 | |
603 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 603 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)(3,14)(4,8)(5,23)(6,11)(7,13)(9,17)(10,20)(12,24)(15,27)(16,30)(18,29)(19,31)(21,28)(22,26)(25,32);(2,8)(3,15)(4,20)(5,6)(9,14)(10,13)(11,26)(12,23)(16,18)(17,32)(19,21)(25,28)(27,30)(29,31);(1,3)(2,9)(4,21)(5,18)(6,17)(7,19)(8,25)(10,30)(11,31)(12,27)(13,29)(14,23)(15,20)(16,22)(24,32)(26,28)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,d^b=di,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=fhi,g^d=gi,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72339664, #72339795, #72339797 |
| maximal normal subgroup of | #72339664, #72339795, #72339797 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60323);