Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152107 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12696 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #60817 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,13,14,7)(4,11,19,9)(5,10)(6,12)(8,18,28,21)(15,23,30,24)(16,25,17,26)(20,32,31,29)(22,27);(1,3)(2,7)(4,17)(5,20)(6,16)(8,23)(9,25)(10,29)(11,13)(12,26)(14,19)(15,22)(18,30)(21,31)(24,27)(28,32);(1,4,6,19)(2,8,12,28)(3,14,16,17)(5,21,22,18)(7,23,26,32)(9,27,11,10)(13,29,25,24)(15,31,20,30)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,e^c=ei,f^c=fi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72339819, #72339835, #72339836, … |
| maximal normal subgroup of | #72339819, #72339835, #72339836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59304);