Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,18,17)(15,20,16,19)(21,22,25,31)(23,29,33,28)(24,35,34,36)(26,32,27,30);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,18)(17,20)(21,23,25,33)(22,28,31,29)(24,26,34,27)(30,36,32,35);(2,4)(7,11)(9,12)(13,16,18,15)(14,19,17,20)(21,24,25,34)(22,29,31,28)(23,26,33,27)(30,36,32,35);(13,17,18,14)(15,20,16,19)(22,30)(24,34)(26,27)(28,36)(29,35)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367651);