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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(10,13)(11,15)(17,18,23,25,22,27,21,28)(19,24,34,39,29,38,31,36)(20,26,35,40,32,42,33,41)(30,37,45,47,43,48,44,46);(1,2,5,6)(3,7,4,8)(10,13)(11,15)(17,19)(18,24)(20,30)(21,31)(22,29)(23,34)(25,39)(26,37)(27,38)(28,36)(32,43)(33,44)(35,45)(40,47)(41,46)(42,48);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(18,25)(19,29)(20,32)(21,23)(24,36)(26,41)(27,28)(37,47)(38,39)(40,42)(44,45)(46,48);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,20,22,32)(18,26,27,42)(19,30,29,43)(21,33,23,35)(24,37,38,48)(25,40,28,41)(31,44,34,45)(36,46,39,47);(9,11,12,15)(10,14,13,16)(18,27)(19,29)(20,32)(25,28)(31,34)(33,35)(37,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7079442);