Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144059 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1677 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,4)(5,6)(7,19,24,35)(8,25)(9,14)(10,17)(11,16)(12,18)(13,21,31,36)(15,27)(20,33,22,30)(23,29,28,34)(26,32);(2,4)(5,7)(6,13)(8,20)(9,22)(10,28)(11,23)(12,24)(14,19)(15,29)(16,33)(17,30)(18,31)(21,26)(25,35)(27,36)(32,34);(6,14)(7,20)(8,26)(9,27)(10,11)(15,17)(16,32)(18,25)(19,29)(21,28)(22,24)(23,36)(30,33)(34,35)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fgi,g^b=gi,d^c=dgi,e^c=eh,f^c=fgi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51255);