Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,30,35)(27,37,38,32)(28,34,44,52)(29,36,31,33)(39,50,42,47)(40,48,41,46)(43,49,45,51)(53,55,54,56);(2,5)(6,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,32)(28,42)(29,40)(30,38)(31,41)(33,48)(34,50)(35,37)(36,46)(39,44)(43,54)(45,53)(47,52)(49,55)(51,56);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,33)(27,38)(29,31)(32,46)(34,49)(35,36)(37,48)(39,42)(43,45)(47,55)(50,56)(51,52);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,44)(26,34,35,52)(27,39,38,42)(29,45,31,43)(32,47,37,50)(33,49,36,51)(40,54,41,53)(46,55,48,56);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(25,29,30,31)(26,33,35,36)(27,40,38,41)(28,43,44,45)(32,48,37,46)(34,51,52,49)(39,53,42,54)(47,55,50,56)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=hi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6397389);