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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1612 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,23,27)(19,28,31,37)(20,25,22,35)(21,26,34,42)(24,36,39,29)(30,43,46,47)(32,40,33,41)(38,45,48,44);(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,44)(33,45)(34,46)(40,47)(41,43)(42,48);(2,5)(6,8)(10,13)(14,16)(17,20,23,22)(18,25,27,35)(19,29,31,36)(21,33,34,32)(24,37,39,28)(26,41,42,40)(30,45,46,44)(38,43,48,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21)(18,26)(19,30)(20,32)(22,33)(23,34)(24,38)(25,40)(27,42)(28,43)(29,44)(31,46)(35,41)(36,45)(37,47)(39,48);(9,11,12,15)(10,14,13,16)(21,34)(26,42)(30,46)(32,33)(38,48)(40,41)(43,47)(44,45)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,e^2=h,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, 6402476);