Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478213 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
877 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 877 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,16,10,15)(7,17,21,28,20,18,25,12)(8,22,24,13,23,30,26,29)(19,33,34,32,36,27,35,31);(1,2)(3,4)(5,6,9,14,11,16,10,15)(7,18,25,28,20,17,21,12)(8,22,26,29,23,30,24,13)(19,27,34,31,36,33,35,32);(2,4)(5,7,10,21,11,20,9,25)(6,12,15,17,16,28,14,18)(8,19,24,35,23,36,26,34)(13,27,30,32,29,33,22,31);(5,8,9,24,11,23,10,26)(6,13,14,30,16,29,15,22)(7,19,21,34,20,36,25,35)(12,27,17,31,28,33,18,32)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=efi,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385410);