Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,30,32)(27,37,38,49)(28,34,44,47)(29,36,31,35)(33,40,46,41)(39,53,42,56)(43,52,45,51)(48,55,50,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(26,32)(27,38)(34,47)(35,36)(39,42)(40,41)(51,52)(54,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,29,40,30,38,31,41)(26,33,35,49,32,46,36,37)(28,42,45,54,44,39,43,55)(34,50,52,53,47,48,51,56);(2,6)(3,8)(11,20)(15,18)(19,21)(23,24)(25,28)(26,34)(27,39)(29,45)(30,44)(31,43)(32,47)(33,48)(35,52)(36,51)(37,53)(38,42)(40,55)(41,54)(46,50)(49,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,29,30,31)(26,35,32,36)(27,40,38,41)(28,43,44,45)(33,49,46,37)(34,51,47,52)(39,54,42,55)(48,53,50,56)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fg,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7242552);