Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1468 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)(13,14)(15,17)(16,18)(19,20)(21,22,24,29,26,30,25,27)(23,31,33,28,32,35,34,36);(2,4)(5,6,8,10)(7,11,12,9)(13,15,16,19)(14,17,18,20)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,7,8,12)(6,9,10,11)(21,23)(22,28)(24,34)(25,33)(26,32)(27,35)(29,31)(30,36);(14,18)(17,20)(23,32)(28,36)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=e,b^2=gh,c^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318067);