Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133181 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #34597 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #2298 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
588 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18)(19,26,28,22)(20,23)(21,25)(24,29)(27,31,32,30);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(18,22)(23,30)(24,31)(25,26);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(18,23,25,24)(22,30,26,31)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fgh,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72293045 |
| maximal normal subgroup of | #72293045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40377);