Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | #1580 |
| prime radical | C2×D8 | C2×C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ #1312 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
148 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 3 generators on 32 points:
(2,6,8,14)(3,10)(4,11)(7,13,15,9)(12,16)(17,18,20,24,21,19,26,27,22,25,29,32,30,28,23,31);(1,2,5,8)(3,7,12,15)(4,13,11,9)(6,10,14,16);(1,3,11,16,5,12,4,10)(2,7,9,6,8,15,13,14)(17,19,20,27,21,25,26,32,22,28,29,31,30,18,23,24)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=efh,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cfi,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286931, #72286988, #72287088, … |
| maximal normal subgroup of | #72286931, #72286988, #72287088, … |
| maximal quotient of | #72323194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30166);