Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1649 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)(17,18,23,27)(19,24,22,37)(20,30,33,38)(21,26,36,44)(25,31,41,28)(29,39,34,42)(32,47,35,48)(40,46,43,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,34,36,29)(25,30,41,38)(26,42,44,39)(32,45,35,46)(40,48,43,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,35,36,32)(24,38,37,30)(26,43,44,40)(29,46,34,45)(39,47,42,48);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,29)(20,32)(22,34)(23,36)(24,39)(25,40)(27,44)(28,45)(30,47)(31,46)(33,35)(37,42)(38,48)(41,43);(2,6)(5,7)(10,14)(13,15)(21,36)(26,44)(29,34)(32,35)(39,42)(40,43)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7208268);