Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485047 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #37483 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 32 points:
(1,2,5,10)(3,12,15,8)(4,7,16,9)(6,11,18,25)(13,27,29,23)(14,22,31,19)(17,20,32,24)(21,28,26,30);(2,7)(3,13)(4,16)(8,21)(9,10)(11,20)(12,26)(14,30)(15,29)(17,32)(19,23)(22,27)(24,25)(28,31);(4,17)(14,28)(16,32)(30,31);(1,3,5,15)(2,8,10,12)(4,14,16,31)(6,13,18,29)(7,19,9,22)(11,23,25,27)(17,28,32,30)(20,21,24,26)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392443);