Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2856 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14757 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1813 conjugacy classes of subgroups; only the 260 normal subgroups are shown.
This group has 1813 conjugacy classes of subgroups — too many to draw. Only its 260 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:
(1,2)(3,15)(4,21)(5,26)(6,13)(7,12)(8,14)(9,18)(10,24)(11,27)(16,28)(17,23)(19,31)(20,30)(22,29)(25,32);(1,3,6,18,8,20,7,19)(2,9,12,15,14,31,13,30)(4,22,23,16,25,11,24,26)(5,17,27,21,28,10,29,32);(1,4,6,24,8,25,7,23)(2,10,12,21,14,17,13,32)(3,16,19,11,20,26,18,22)(5,15,27,9,28,30,29,31);(1,5)(2,11)(3,17)(4,15)(6,27)(7,29)(8,28)(9,24)(10,20)(12,26)(13,16)(14,22)(18,21)(19,32)(23,31)(25,30)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305099);