Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
888 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 170 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:
(1,2,4,3)(5,6)(7,18)(8,22)(9,27)(10,16)(11,15)(12,17)(13,20)(14,25)(19,34)(21,30)(23,36)(24,33)(26,32)(28,29)(31,35);(1,2)(3,4)(5,7,12,9)(6,13,17,30)(8,23,26,35)(10,21,11,20)(14,31,33,36)(15,27,16,18)(19,25,28,24)(22,29,32,34);(1,3,4,2)(5,8)(6,14)(7,19)(9,28)(10,25)(11,24)(12,26)(13,29)(15,22)(16,32)(17,33)(18,31)(20,35)(21,23)(27,36)(30,34);(2,3)(5,9,12,7)(6,13,17,30)(8,23,26,35)(10,20,11,21)(14,31,33,36)(15,27,16,18)(19,25,28,24)(22,29,32,34)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384525);