Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(2,4)(5,6,7,9,8,11,12,10)(14,18,20,23)(15,22)(16,25)(19,24)(21,28,27,26);(6,9,11,10)(13,14,19,26,25,27,24,18,17,20,22,28,16,21,15,23);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72216171, #72216172, #72216174, … |
| maximal normal subgroup of | #72216171, #72216172, #72216174, … |
| maximal quotient of | #72216171, #72225158, #72225171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47722);