Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #24364 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,36)(21,27)(22,29)(23,28)(24,30)(26,37)(31,45)(32,40)(33,42)(34,41)(35,43)(38,44)(39,46)(47,48);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,24,35)(18,25,30,43)(20,31,21,32)(22,33,23,34)(26,39,44,48)(27,40,36,45)(28,41,29,42)(37,46,38,47);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,24)(22,37)(23,38)(25,39)(27,29)(28,36)(30,44)(32,35)(33,46)(34,47)(40,42)(41,45)(43,48);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,21,24,20)(18,27,30,36)(19,32,35,31)(22,37,23,38)(25,40,43,45)(26,28,44,29)(33,46,34,47)(39,41,48,42);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,21,38)(25,41,43,42)(26,36,44,27)(31,46,32,47)(39,45,48,40)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6904230);