Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454389 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2815 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17057 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
802 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 184 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:
(1,2,5,12,7,13,6,11)(3,14,16,8,18,26,17,10)(4,19,22,9,24,29,23,30)(15,31,20,28,32,25,21,27);(1,3,5,16,7,18,6,17)(2,8,11,14,13,10,12,26)(4,20,22,32,24,21,23,15)(9,27,19,25,30,28,29,31);(1,4,6,22,7,24,5,23)(2,9,12,19,13,30,11,29)(3,15,17,20,18,32,16,21)(8,25,26,28,10,31,14,27);(1,3,5,16,7,18,6,17)(2,10,11,26,13,8,12,14)(4,21,22,15,24,20,23,32)(9,28,19,31,30,27,29,25)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382703);