Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,26,31)(23,32,35,36)(24,30,25,29)(27,34,28,33);(5,7,9,12)(6,10,8,11)(13,15)(14,17)(16,19)(18,20);(5,8,9,6)(7,11,12,10)(15,19)(17,20)(21,23,25,34,26,35,24,33)(22,27,30,36,31,28,29,32);(2,4)(7,12)(10,11)(21,23,24,33,26,35,25,34)(22,28,29,36,31,27,30,32)Polycyclic presentation:
2^9|a^2=egi,b^2=e,c^2=efi,d^2=f,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=de,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344291);