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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,30,41)(20,32,21,26)(22,36,37,28)(25,35,43,34)(29,39,33,48)(31,46,45,40)(38,47,42,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,29)(22,31)(23,30)(25,42)(26,39)(27,41)(28,40)(32,48)(34,47)(35,44)(36,46)(37,45)(38,43);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,23)(22,35)(24,38)(26,28)(27,43)(30,33)(31,47)(32,36)(34,37)(39,46)(40,48)(41,42)(44,45);(3,8)(5,7)(11,16)(13,15)(17,21,23,20)(18,26,27,32)(19,29,30,33)(22,35,37,34)(24,39,41,48)(25,36,43,28)(31,44,45,47)(38,40,42,46);(10,14)(13,15)(18,27)(19,30)(20,21)(28,36)(31,45)(34,35)(38,42)(39,48)Polycyclic presentation:
2^9|a^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,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, 6901737);