Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142232 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
641 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 641 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,20)(15,25)(16,27)(17,18)(21,23)(22,28)(24,26);(9,12)(13,14,19,24)(15,26,27,20)(16,28,25,21)(17,23,18,22);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,16,19,27,17,25)(14,21,23,20,24,28,22,26)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,g^b=gi,d^c=dh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49497);