Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324436 |
| 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 | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,50)(38,46)(40,51)(41,47)(42,49)(43,48)(45,52)(53,55)(54,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23);(1,3)(2,5)(4,8)(6,7)(25,27,29,38,31,40,30,39)(26,32,44,51,35,46,34,36)(28,37,43,54,42,53,41,52)(33,45,47,55,49,56,48,50);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,28,31,42)(26,33,35,49)(27,37,40,53)(29,43,30,41)(32,45,46,56)(34,48,44,47)(36,50,51,55)(38,54,39,52);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,41,42,43)(32,36,46,51)(33,47,49,48)(37,52,53,54)(45,55,56,50)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=i,e^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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7240375);