Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1265 conjugacy classes of subgroups — too many to draw. Only its 178 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)(5,6,9,13,10,14,15,23)(7,11,17,26,18,27,31,33)(8,12,21,19,22,29,24,30)(16,25,20,28,32,35,34,36);(2,3)(5,6)(7,11)(8,19)(9,23)(10,14)(12,21)(13,15)(16,28)(17,33)(18,27)(20,25)(22,30)(24,29)(26,31)(32,36)(34,35);(1,3,4,2)(5,7,15,31,10,18,9,17)(6,11,23,33,14,27,13,26)(8,20,24,16,22,34,21,32)(12,28,30,25,29,36,19,35);(2,3)(5,8)(6,12)(7,16)(9,24)(10,22)(11,25)(13,30)(14,29)(15,21)(17,34)(18,32)(19,23)(20,31)(26,36)(27,35)(28,33)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383686);