Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1316 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | #379 | C23 |
| socle | C2 | C2×#1316 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1086 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 32 points:
(1,2,5,10)(3,11,13,25,14,7,17,20)(4,8,15,22)(6,9,18,23)(12,24,26,32,27,19,28,30)(16,21,29,31);(1,3)(2,7)(4,12)(5,17)(6,14)(8,19)(9,11)(10,20)(13,18)(15,28)(16,27)(21,24)(22,30)(23,25)(26,29)(31,32);(1,4,6,16)(2,8,9,21)(3,12,14,27)(5,15,18,29)(7,19,11,24)(10,22,23,31)(13,26,17,28)(20,30,25,32)Polycyclic presentation:
2^9|a^2=e,c^2=i,e^2=g,b^a=bd,d^a=df,f^a=fh,h^a=hi,e^b=ef,g^b=gi,e^d=eh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338317, #72338379, #72338380 |
| maximal normal subgroup of | #72338317, #72338379, #72338380 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56724);