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 | #45091 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,32)(30,41)(31,43)(33,38)(34,42)(35,37)(36,40)(39,46)(44,48)(45,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,33)(27,28)(32,35)(34,36)(39,47)(43,48);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,23,32,22,30,21,35)(18,25,28,29,24,37,27,41)(20,31,34,46,33,44,36,45)(26,39,40,43,38,47,42,48);(3,8)(6,7)(11,16)(13,15)(17,20,22,33)(18,26,24,38)(19,31,30,44)(21,36,23,34)(25,39,37,47)(27,42,28,40)(29,43,41,48)(32,46,35,45);(10,14)(13,15)(17,21,22,23)(18,27,24,28)(19,32,30,35)(20,34,33,36)(25,29,37,41)(26,40,38,42)(31,45,44,46)(39,48,47,43)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6927514);