Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1108 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)(8,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,29)(24,28)(25,27)(26,34)(30,33)(31,36)(32,35);(5,7,10,6)(8,12,9,11)(13,15,16,20)(14,17,18,19)(22,27)(23,31)(24,32)(30,34);(5,8,10,9)(6,11,7,12)(15,20)(17,19)(21,23,26,24,25,31,33,32)(22,28,30,29,27,35,34,36);(5,9,10,8)(6,11,7,12)(14,18)(15,20)(21,24,26,31,25,32,33,23)(22,29,30,35,27,36,34,28)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=efg,d^2=efg,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295296);