Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1635 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,29,42)(20,33,34,25)(21,26,22,37)(24,31,38,32)(30,45,44,43)(35,41,36,40)(39,46,48,47);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(19,29)(20,34)(24,38)(25,33)(28,42)(31,32)(35,36)(40,41);(11,15)(13,16)(17,19,23,29)(18,24,27,38)(20,30,34,44)(21,32,22,31)(25,39,33,48)(26,42,37,28)(35,47,36,46)(40,43,41,45);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,39)(26,41)(27,33)(28,43)(29,44)(31,47)(32,46)(37,40)(38,48)(42,45);(2,5)(6,8)(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,35,34,36)(24,28,38,42)(25,40,33,41)(30,46,44,47)(39,45,48,43)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^c=ei,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, 7108927);