Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188035 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2370 conjugacy classes of subgroups; only the 358 normal subgroups are shown.
This group has 2370 conjugacy classes of subgroups — too many to draw. Only its 358 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,13,15,14,7,12,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3,6,14)(2,7,9,11)(4,13,5,12)(8,16,10,15)(20,28)(23,30)(26,32)(29,31);(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,11,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268177, #72268179, #72268191, … |
| maximal normal subgroup of | #72268177, #72268179, #72268191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95357);