Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| 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 | |
518 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 518 conjugacy classes of subgroups — too many to draw. Only its 134 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)(6,9,11,12)(13,14,15,20,16,21,23,18,17,22,24,19,25,27,28,26);(5,6,10,9,8,11,7,12)(14,18,22,26)(15,23)(16,25)(19,21,20,27)(24,28);(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,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216299, #72216334, #72216338, … |
| maximal normal subgroup of | #72216299, #72216334, #72216338, … |
| maximal quotient of | #72216297, #72216299, #72225357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48042);