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 | |
1040 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)(21,22,25,28,26,30,24,29)(23,27,32,34,33,36,31,35);(5,7)(6,8)(9,12)(10,11)(13,14,16,18)(15,19,20,17)(21,23,26,33)(22,27,30,36)(24,31,25,32)(28,34,29,35);(2,4)(5,8,10,9)(6,11,12,7)(13,15,16,20)(14,17,18,19)(22,28)(24,25)(27,34)(29,30)(31,32)(35,36);(2,4)(5,9,10,8)(6,11,12,7)(22,29)(24,25)(27,35)(28,30)(31,32)(34,36)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=eg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294345);