Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(2,3)(5,6,9,11,8,7,10,12)(13,14,16,22,18,23,15,20)(17,26,25,28,27,19,24,21);(1,2,4,3)(5,7,9,12,8,6,10,11)(13,15,18,16)(17,25,27,24)(19,26)(21,28);(1,3,4,2)(6,7)(11,12)(14,19)(15,16)(17,25)(20,21)(22,28)(23,26)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=dg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318512, #72318530, #72318584, … |
| maximal normal subgroup of | #72318512, #72318530, #72318584, … |
| maximal quotient of | #72322473 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47342);