Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
422 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,11)(8,19)(12,28)(13,27)(14,31)(15,20)(16,33)(17,23)(22,24)(26,29)(30,34)(32,36);(1,3,4,2)(5,6)(7,12)(8,20)(9,14)(10,15)(11,19)(13,25)(16,34)(17,29)(18,30)(21,36)(22,31)(23,32)(24,27)(26,35)(28,33);(5,7,10,18)(6,12,15,30)(8,21,19,35)(9,23,25,17)(11,26,20,36)(13,29,14,32)(16,24,33,22)(27,28,31,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gh,d^b=di,e^b=eghi,h^b=hi,d^c=dg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,g^e=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59659);