Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #29675 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1465 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)(3,7)(4,6)(5,8)(17,18,23,25,21,27,22,28)(19,29,36,40,31,24,32,38)(20,33,37,42,34,26,35,41)(30,39,46,47,44,43,45,48);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,24)(20,30)(21,31)(22,32)(23,36)(25,38)(26,39)(27,29)(28,40)(33,43)(34,44)(35,45)(37,46)(41,47)(42,48);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(18,25)(20,34)(22,23)(24,38)(26,42)(27,28)(29,40)(30,44)(32,36)(33,41)(39,48)(43,47);(2,6)(5,7)(10,13)(14,16)(17,20)(18,26)(19,30)(21,34)(22,35)(23,37)(24,39)(25,41)(27,33)(28,42)(29,43)(31,44)(32,45)(36,46)(38,47)(40,48);(9,11,12,15)(10,14,13,16)(18,27)(19,31)(20,34)(25,28)(32,36)(35,37)(39,43)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6873630);