Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636940 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1445 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)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,21,27)(20,33,37,42)(22,29,23,28)(26,35,43,36)(31,45,34,47)(32,44,38,40)(39,48,41,46);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,34)(21,24)(22,38)(23,32)(26,41)(27,30)(28,44)(29,40)(31,37)(33,47)(35,46)(36,48)(39,43)(42,45);(11,15)(13,16)(17,20,23,36,24,37,22,35)(18,26,29,33,30,43,28,42)(19,31,32,46,21,34,38,48)(25,39,40,45,27,41,44,47);(2,6)(3,8)(11,15)(13,16)(17,21,24,19)(18,27,30,25)(20,34,37,31)(22,32,23,38)(26,41,43,39)(28,40,29,44)(33,47,42,45)(35,46,36,48);(2,6)(5,7)(10,14)(13,16)(17,22,24,23)(18,28,30,29)(19,32,21,38)(20,35,37,36)(25,40,27,44)(26,42,43,33)(31,46,34,48)(39,45,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6544459);