Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1524 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)(3,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,28,31,24)(20,32,35,25)(21,36,23,26)(29,39,47,45)(30,40,37,41)(33,42,46,48)(34,43,38,44);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,37)(25,42)(26,40)(27,28)(29,34)(32,48)(35,46)(36,41)(38,47)(39,44)(43,45);(10,13)(14,16)(17,20,22,35)(18,25,27,32)(19,29,31,47)(21,34,23,38)(24,39,28,45)(26,43,36,44)(30,46,37,33)(40,42,41,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,34,35,38)(24,40,28,41)(25,43,32,44)(29,46,47,33)(39,42,45,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,27)(19,30)(20,35)(21,23)(24,40)(28,41)(29,33)(31,37)(39,48)(42,45)(43,44)(46,47)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fgi,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,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6640471);