Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #61436 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
761 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 761 conjugacy classes of subgroups — too many to draw. Only its 240 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,23,25,32)(19,21)(20,27)(24,28,30,31)(26,29);(17,18,22,25)(19,24,21,30)(20,28,29,31)(23,27,32,26);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,21)(18,24,25,30)(20,26,29,27)(23,31,32,28);(3,9)(6,12)(10,15)(13,16)(17,19,22,21)(18,24,25,30)(20,26,29,27)(23,31,32,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216809, #72216897, #72216933, … |
| maximal normal subgroup of | #72216809, #72216897, #72216933, … |
| maximal quotient of | #72286006, #72286291, #72343740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388626);