Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1408 |
| commutator | #376 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| socle | C2 | #7402 |
| prime radical | #10012 | C2 |
| LCS | G ▷ #376 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 32 points:
(1,2,4,8)(3,12,13,23)(5,9,19,25)(6,11,17,26)(7,16,21,15)(10,18,28,20)(14,27,29,24)(22,30,32,31);(2,7)(8,23)(9,27)(10,12)(11,22)(13,29)(16,30)(21,25)(24,26)(28,32)Polycyclic presentation:
2^9|a^2=d,c^2=f,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,d^c=dghi,e^c=ei,e^d=ehi,f^d=fi,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72338345, #72338411, #72339807 |
| maximal normal subgroup of | #72338345, #72338411, #72339807 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1830);