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 | #45093 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(19,27)(20,25)(21,36)(22,23)(24,32)(26,29)(28,40)(30,43)(31,37)(33,38)(34,42)(35,41)(39,45)(44,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,33)(26,36)(31,32)(34,35)(39,47)(43,48);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,29,31,22,28,21,32)(18,24,36,40,23,37,26,27)(20,30,34,46,33,44,35,45)(25,39,41,48,38,47,42,43);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,33)(23,38)(24,39)(26,42)(27,43)(28,44)(29,34)(31,46)(32,45)(36,41)(37,47)(40,48);(10,14)(13,15)(17,21,22,29)(18,26,23,36)(19,31,28,32)(20,34,33,35)(24,40,37,27)(25,41,38,42)(30,45,44,46)(39,43,47,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,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, 6927508);