Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1356 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,30,35,29,34,44,32)(27,36,39,46,38,31,45,48)(28,40,43,49,42,33,50,52)(37,47,53,56,41,51,54,55);(2,6)(3,4)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27,29,38)(26,31,34,36)(28,41,42,37)(30,39,44,45)(32,46,35,48)(33,51,40,47)(43,54,50,53)(49,56,52,55);(1,3,5,4)(2,7,6,8)(26,32)(30,44)(31,46)(33,49)(34,35)(36,48)(39,45)(40,52)(43,50)(47,55)(51,56)(53,54);(2,6)(7,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,29,42)(26,33,34,40)(27,37,38,41)(30,43,44,50)(31,47,36,51)(32,49,35,52)(39,53,45,54)(46,55,48,56);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(26,34)(28,42)(31,36)(32,35)(37,41)(43,50)(46,48)(53,54)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,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, 7138414);