Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45134 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(10,13)(11,15)(17,18,23,27)(19,24,22,37)(20,30,32,46)(21,26,35,42)(25,28,40,33)(29,38,36,48)(31,45,34,47)(39,43,41,44);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,23,22)(18,24,27,37)(20,28,32,33)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,33,23,32,19,28)(18,25,24,30,27,40,37,46)(21,34,36,43,35,31,29,44)(26,41,38,47,42,39,48,45);(2,5)(6,8)(10,13)(14,16)(17,21)(18,26)(19,29)(20,31)(22,36)(23,35)(24,38)(25,39)(27,42)(28,43)(30,45)(32,34)(33,44)(37,48)(40,41)(46,47);(9,11,12,15)(10,14,13,16)(20,32)(21,35)(25,40)(26,42)(28,33)(29,36)(30,46)(38,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,e^2=h,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102321);