Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,22)(23,33)(24,36)(25,28)(26,32)(27,31)(29,35)(30,34);(1,2,3,4)(6,9)(10,11)(13,15,17,16)(14,18,20,19)(22,28)(23,34)(24,35)(31,32);(2,4)(5,7,8,12)(6,10,9,11)(13,15,17,16)(14,19,20,18)(21,23,27,24,25,34,26,35)(22,29,32,30,28,36,31,33);(7,12)(10,11)(13,16,17,15)(14,19,20,18)(21,24,27,34,25,35,26,23)(22,30,32,36,28,33,31,29)Polycyclic presentation:
2^9|a^2=ei,b^2=efgh,c^2=fi,d^2=ef,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^b=di,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380938);