Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #39131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,33)(28,35)(29,36)(30,34)(31,39)(32,38)(37,43)(40,44)(41,45)(42,47)(46,48);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(17,19,22,29)(18,23,27,36)(20,28,31,42)(21,30,33,43)(24,34,26,37)(25,35,39,47)(32,41,44,48)(38,45,40,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(18,24)(21,33)(23,34)(25,38)(26,27)(30,43)(32,44)(35,45)(36,37)(39,40)(41,48)(46,47);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(29,42)(30,41)(33,44)(34,45)(36,47)(37,46)(43,48);(1,4,5,3)(2,6,7,8)(19,29)(20,31)(23,36)(25,39)(30,43)(32,44)(34,37)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6773933);