Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232273 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1486 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1486 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,14)(7,15,18,19)(8,12,20,17)(9,16,10,13)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,35)(34,36);(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,20)(15,16)(21,23,25,30)(22,26,28,33)(24,29,31,36)(27,34,35,32);(2,4)(5,8,11,20)(6,12,14,17)(7,16,18,13)(9,15,10,19)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,36)(33,34);(5,9,11,10)(6,13,14,16)(7,12,18,17)(8,19,20,15)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,d^c=dh,e^c=ei,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139478);