Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503222 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25771 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1133 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1133 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,16)(5,10)(9,22)(11,21)(13,29)(14,31)(15,28)(17,32)(18,25)(20,24)(23,27)(26,30);(1,2,5,8,7,4,10,16)(3,9,14,20,15,13,23,30)(6,17,18,32,19,11,25,21)(12,24,27,22,28,26,31,29);(1,3,6,12)(2,9,11,24)(4,13,17,26)(5,14,18,27)(7,15,19,28)(8,20,21,22)(10,23,25,31)(16,30,32,29);(1,4,5,16,7,2,10,8)(3,13,14,30,15,9,23,20)(6,17,18,32,19,11,25,21)(12,26,27,29,28,24,31,22)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237523, #72237525, #72237546, … |
| maximal normal subgroup of | #72237523, #72237525, #72237546, … |
| maximal quotient of | #72312331, #72312333, #72312342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410418);