Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6,8,11)(7,9,12,10)(13,14,18,22)(15,24,20,17)(16,21,27,26)(19,25,28,23);(6,9,11,10)(13,15,16,25,19,26,24,14,18,20,27,23,28,21,17,22);(1,3,4,2)(13,16,19,24,18,27,28,17)(14,20,23,21,22,15,25,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=egi,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343469, #72343471, #72343478, … |
| maximal normal subgroup of | #72343469, #72343471, #72343478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35800);