Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1643 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)(17,18,23,29)(19,31,34,25)(20,36,21,27)(22,30,24,28)(26,39,44,37)(32,45,38,41)(33,42,35,43)(40,47,46,48);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,32)(21,38)(22,33)(23,34)(24,35)(26,40)(27,45)(28,42)(29,31)(30,43)(36,41)(37,47)(39,48)(44,46);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,32)(21,23)(22,37)(24,39)(25,40)(27,30)(28,36)(29,44)(31,46)(33,47)(34,38)(35,48)(41,42)(43,45);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,23,20)(18,27,29,36)(19,32,34,38)(22,37,24,39)(25,41,31,45)(26,28,44,30)(33,48,35,47)(40,43,46,42);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,33,34,35)(20,37,21,39)(25,42,31,43)(26,36,44,27)(32,47,38,48)(40,41,46,45)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7143522);