Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
1679 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,26)(21,35)(22,25)(23,27)(28,41)(29,37)(30,38)(31,36)(32,39)(33,45)(34,40)(42,47)(43,46)(44,48);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,29)(21,31)(22,30)(23,28)(25,38)(26,37)(27,41)(32,44)(33,43)(34,42)(35,36)(39,48)(40,47)(45,46);(9,11,13,12)(10,14,15,16)(18,25)(19,28)(20,32)(21,23)(24,36)(29,43)(30,31)(37,47)(38,41)(40,45)(42,44)(46,48);(2,6)(5,7)(17,20,22,32)(18,26,25,39)(19,29,30,44)(21,33,23,34)(24,37,38,48)(27,40,35,45)(28,42,31,43)(36,46,41,47);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21)(18,27)(20,33)(22,23)(24,38)(25,35)(26,40)(28,31)(32,34)(37,48)(39,45)(42,43)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6627879);