Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485043 |
| 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 | #23984 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,11)(4,16)(5,8)(6,10)(7,14)(9,19)(12,20)(13,25)(15,23)(17,22)(18,29)(21,28)(24,27)(26,32)(30,31);(1,3,12,26)(2,7,20,31)(4,17,27,28)(5,15,6,13)(8,22,10,21)(9,23,29,25)(11,24,32,16)(14,18,30,19);(3,13)(7,21)(9,24)(14,28)(15,26)(16,29)(17,30)(22,31);(2,8)(7,22)(9,16)(10,20)(11,23)(21,31)(24,29)(25,32)Polycyclic presentation:
2^9|b^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72241080, #72241095, #72278573, … |
| maximal normal subgroup of | #72241080, #72241095, #72278573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392278);