Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26,30,34)(27,36,29,32)(28,39,41,55)(31,45,44,35)(33,48,49,42)(37,52,40,54)(38,47,43,51)(46,53,50,56);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27)(26,32)(28,37)(29,30)(31,38)(33,46)(34,36)(35,47)(39,54)(40,41)(42,53)(43,44)(45,51)(48,56)(49,50)(52,55);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,19)(18,23)(25,28,31,42,30,41,44,48)(26,33,35,39,34,49,45,55)(27,37,38,53,29,40,43,56)(32,46,47,54,36,50,51,52);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,29,30,27)(26,32,34,36)(28,40,41,37)(31,43,44,38)(33,46,49,50)(35,47,45,51)(39,54,55,52)(42,56,48,53);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(27,29)(28,41)(32,36)(33,49)(38,43)(39,55)(42,48)(47,51)Polycyclic presentation:
2^9|a^2=i,c^2=fghi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6406297);