Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #31059 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(17,18,22,27)(19,28,23,24)(20,31,33,42)(21,26,35,41)(25,36,39,29)(30,40,34,37)(32,44,46,47)(38,45,48,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,36)(21,34,35,30)(25,31,39,42)(26,40,41,37)(32,45,46,43)(38,47,48,44);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20,22,33)(18,25,27,39)(19,29,23,36)(21,32,35,46)(24,31,28,42)(26,38,41,48)(30,43,34,45)(37,44,40,47);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21,22,35)(18,26,27,41)(19,30,23,34)(20,32,33,46)(24,37,28,40)(25,38,39,48)(29,43,36,45)(31,44,42,47);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(21,35)(26,41)(30,34)(32,46)(37,40)(38,48)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7185451);