Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35659 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(16,30)(24,35)(29,36)(31,34);(1,2)(3,4)(5,6,10,14)(7,15,18,28)(8,12,19,26)(9,21,23,13)(11,22,25,17)(16,29,30,36)(20,33,32,27)(24,34,35,31);(1,3,4,2)(5,7)(6,11)(8,16)(9,22)(10,18)(12,24)(13,28)(14,25)(15,21)(17,23)(19,30)(20,34)(26,35)(27,36)(29,33)(31,32);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(25,35)(28,36)Polycyclic presentation:
2^9|b^2=i,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287045, #72287190, #72287194, … |
| maximal normal subgroup of | #72287045, #72287190, #72287194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301758);