Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93181 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C23×C4 | C42 |
| Frattini, prime radical, cosocle | C22×#120 | C22 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
767 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 767 conjugacy classes of subgroups — too many to draw. Only its 40 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 — 2 generators on 32 points:
(1,2,4,10)(3,14,16,27)(5,21,7,13)(6,11,20,9)(8,24,25,19)(12,23,30,22)(15,29,18,26)(17,31,32,28);(1,3,5,15,7,17,4,19)(2,8,10,26,13,29,21,25)(6,18,23,16,22,24,20,32)(9,14,12,27,30,31,11,28)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,e^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,f^b=fi,g^b=ghi,d^c=dh,e^c=ehi,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72227479, #72227524, #72234506, … |
| maximal normal subgroup of | #72227479, #72227524, #72234506, … |
| maximal quotient of | #72227478, #72236026, #72237023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378);