Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 118 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)(5,6,10,11,9,7,8,12)(14,18,20,22)(15,19)(16,24)(21,26,28,25)(23,27);(5,7,10,12,9,6,8,11)(13,14,19,26,24,28,15,22,17,20,27,25,16,21,23,18);(1,2,3,4)(5,8,9,10)(6,11,7,12)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318714, #72318723, #72318742, … |
| maximal normal subgroup of | #72318714, #72318723, #72318742, … |
| maximal quotient of | #72322542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47770);