Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(5,6,8,11,9,7,10,12)(13,14,16,22,17,23,26,21,18,24,28,19,27,20,15,25);(1,2,3,4)(14,19)(15,16)(17,27)(20,25)(21,23)(22,24)(26,28);(5,7,8,12,9,6,10,11)(14,20,24,23)(15,26)(16,28)(19,25,22,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12067, C2×#129165, #72223644, … |
| maximal normal subgroup of | C4×#12067, C2×#129165, #72223644, … |
| maximal quotient of | #72223570, C2×#129165, #72223644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36361);