Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 | |
1284 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)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,28)(25,30)(26,29)(31,34)(32,35)(33,36);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,31,33,26,24,25,32)(22,27,34,36,29,28,30,35);(2,4)(7,11)(10,12)(21,24,26,23)(22,28,29,27)(25,32,31,33)(30,35,34,36);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(22,29)(27,28)(30,34)(35,36)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378701);