Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152133 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12707 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #535 | C23 |
| prime radical | #61630 | C2 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3484 ◁ G | |
| LECS | G ▷ #535 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
739 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 739 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,14,26)(4,15,16,31)(5,18,20,9)(6,10)(7,12,21,28)(8,17)(13,24,19,27)(22,29,23,30)(25,32);(1,3)(2,7)(4,12)(5,19)(6,13)(8,24)(9,21)(10,23)(11,25)(14,20)(15,27)(16,30)(17,28)(18,22)(26,31)(29,32);(8,25)(11,24)(15,31)(26,27)Polycyclic presentation:
2^9|a^2=f,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72216447, C2×#152133, #72292695 |
| maximal normal subgroup of | #72216447, C2×#152133, #72292695 |
| maximal quotient of | C2×#152133, #72292954, #72292975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59329);