Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | #30512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1571 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,33)(28,35)(29,34)(30,36)(31,40)(32,38)(37,43)(39,44)(41,47)(42,45)(46,48);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,26,36)(20,28,31,41)(21,29,33,43)(24,34,27,37)(25,35,40,47)(32,42,44,48)(38,45,39,46);(11,15)(13,16)(18,24)(21,33)(23,34)(25,38)(26,27)(29,43)(32,44)(35,45)(36,37)(39,40)(42,48)(46,47);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(29,42)(30,41)(33,44)(34,45)(36,47)(37,46)(43,48);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(18,26)(20,31)(23,36)(24,27)(28,41)(32,44)(34,37)(42,48)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7113434);