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 | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1548 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)(17,18,23,28)(19,29,32,42)(20,25,34,40)(21,27,22,26)(24,36,38,31)(30,43,46,47)(33,41,35,39)(37,44,48,45);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)(17,19,23,32)(18,24,28,38)(20,30,34,46)(21,31,22,36)(25,37,40,48)(26,29,27,42)(33,44,35,45)(39,47,41,43);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,37)(26,41)(27,39)(28,40)(29,43)(31,45)(32,46)(36,44)(38,48)(42,47);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,34,35)(24,29,38,42)(25,39,40,41)(30,44,46,45)(37,47,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=fi,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, 7113373);