Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1014 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)(7,9)(8,11)(10,12)(14,17)(15,19)(21,22)(23,26)(24,28)(25,27)(29,33)(30,35)(31,34)(32,36);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,25,31)(22,26,27,34)(24,29,32,30)(28,33,36,35);(5,8,10,9)(6,11,12,7)(13,15,16,19)(14,18,17,20)(23,29)(24,32)(26,33)(28,36)(30,31)(34,35);(5,9,10,8)(6,11,12,7)(14,17)(18,20)(22,27)(26,34)(28,36)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292114);