Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,24)(19,27,30,40)(20,25,31,38)(21,33,35,26)(23,34,36,29)(28,41,42,48)(32,45,44,39)(37,46,47,43);(11,12)(14,15)(17,19,22,30)(18,23,24,36)(20,28,31,42)(21,34,35,29)(25,37,38,47)(26,40,33,27)(32,46,44,43)(39,48,45,41);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(18,24)(20,31)(23,36)(26,33)(27,40)(28,42)(32,44)(43,46);(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,28)(21,32)(22,31)(23,37)(24,38)(26,39)(27,41)(29,43)(30,42)(33,45)(34,46)(35,44)(36,47)(40,48);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(20,31)(25,38)(28,42)(32,44)(37,47)(39,45)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=ghi,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396270);