Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 ◁ #2296 ◁ 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)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,34)(27,35,37,51)(28,39,41,32)(29,43,44,33)(31,45,47,38)(36,52,40,50)(42,49,53,55)(46,54,48,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,38,30,37,44,45)(26,31,33,35,34,47,43,51)(28,40,42,56,41,36,53,54)(32,48,49,52,39,46,55,50);(2,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,32)(27,36)(29,42)(30,41)(31,46)(33,49)(34,39)(35,50)(37,40)(38,54)(43,55)(44,53)(45,56)(47,48)(51,52);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(28,41)(32,39)(36,40)(42,53)(46,48)(49,55)(50,52)(54,56);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(27,37)(31,47)(35,51)(36,40)(38,45)(46,48)(50,52)(54,56)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=gi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472870);