Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343554 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62835 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,13)(4,9)(5,11)(6,10)(7,12)(8,14)(15,16)(17,18,21,26,22,25,34,39)(19,27,29,40,30,23,33,36)(20,24,32,38,31,37,44,47)(28,41,43,48,42,35,45,46);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(17,19,22,30)(18,23,25,27)(20,28,31,42)(21,33,34,29)(24,35,37,41)(26,40,39,36)(32,45,44,43)(38,48,47,46);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,20,22,31)(18,24,25,37)(19,28,30,42)(21,32,34,44)(23,35,27,41)(26,38,39,47)(29,43,33,45)(36,46,40,48);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(18,25)(23,27)(24,37)(26,39)(35,41)(36,40)(38,47)(46,48);(1,5,7,6)(2,11,12,10)(3,14,4,15)(8,9,16,13)(20,31)(24,37)(28,42)(32,44)(35,41)(38,47)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,f^2=i,g^2=hi,b^a=bf,d^a=dh,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251444);