Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153129 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(2,3)(5,6,10,16)(7,11,19,27)(8,12,21,13)(9,14,24,15)(17,25,33,36)(18,32,23,26)(20,28,34,31)(22,29,35,30);(1,2)(3,4)(5,7)(6,11)(8,20)(9,23)(10,19)(12,28)(13,31)(14,26)(15,32)(16,27)(17,22)(18,24)(21,34)(25,29)(30,36)(33,35);(1,3,4,2)(6,12)(8,21)(11,25)(13,16)(14,29)(15,30)(18,23)(20,34)(22,35)(26,28)(27,36)(31,32)Polycyclic presentation:
2^9|a^2=i,c^2=d,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,f^a=fgh,g^a=gi,c^b=cf,e^b=eg,g^b=gi,h^b=hi,e^c=ehi,g^c=gh,f^e=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60486);