Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
500 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(6,9)(7,10)(11,12)(14,21)(15,25)(16,17)(18,19)(22,24)(23,27)(26,28);(2,4)(5,6,10,12,8,11,7,9)(13,14,17,22,18,23,26,15,20,25,28,27,19,24,16,21);(13,15,26,24,19,22,17,25,20,21,16,23,18,27,28,14)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ehi,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216165, #72216264, #72216271, … |
| maximal normal subgroup of | #72216165, #72216264, #72216271, … |
| maximal quotient of | #72216196, #72216244, #72216324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48877);