Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,38)(35,43)(37,48)(39,46)(40,45)(41,49)(42,47)(44,52)(50,54)(51,56)(53,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,38)(26,31,34,36)(28,40,41,37)(30,39,43,52)(32,44,35,46)(33,48,49,45)(42,53,54,51)(47,56,50,55);(1,3,4,8)(2,5,6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,43)(31,44)(33,47)(34,35)(36,46)(39,52)(42,54)(45,55)(48,56)(49,50)(51,53);(3,8)(5,7)(11,19)(14,17)(18,24)(21,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,47)(34,49)(35,50)(36,48)(38,40)(39,51)(43,54)(44,55)(46,56)(52,53);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)(26,34)(28,41)(31,36)(32,35)(37,40)(42,54)(44,46)(51,53)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7247336);