Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153117 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#2196 | C2 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
779 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 779 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,13)(7,10,16,20)(8,11,17,12)(14,18,15,19);(10,18)(14,15)(19,20);(2,4)(5,7)(6,10)(8,14)(9,16)(11,18)(12,19)(13,20)(15,17)Polycyclic presentation:
2^9|a^2=i,d^2=h,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,g^c=gi,h^c=hi,f^d=fh,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72250904, #72279813, #72279833, … |
| maximal normal subgroup of | #72250904, #72279813, #72279833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60333);