Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2785 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17223 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
715 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(4,14,17,18)(7,20)(8,23,15,26)(11,27,16,29)(21,31,24,32)(22,30);(1,2)(3,7)(4,15,17,8)(5,10)(6,9)(11,21,16,24)(12,20)(13,22)(14,26,18,23)(19,25)(27,32,29,31)(28,30);(1,3)(2,7)(4,16,17,11)(5,13)(6,12)(8,24,15,21)(9,20)(10,22)(14,29,18,27)(19,28)(23,31,26,32)(25,30);(1,4)(2,8)(3,11)(5,18)(6,17)(7,21)(9,15)(10,26)(12,16)(13,29)(14,19)(20,24)(22,31)(23,25)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386855);