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 | |
346 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 346 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:
(1,2)(3,4)(5,6,8,11,9,7,10,12)(13,14,17,22,19,24,26,23)(15,16,20,27,21,18,25,28);(2,4)(13,15,18,23,17,25,28,24,19,21,16,22,26,20,27,14);(5,7,8,12,9,6,10,11)(13,16,26,28,19,18,17,27)(14,20,23,15,24,25,22,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=cg,e^b=ei,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318782, #72318786, #72318833, … |
| maximal normal subgroup of | #72318782, #72318786, #72318833, … |
| maximal quotient of | #72322517, #72322531 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35885);