Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,35)(28,33)(29,31)(30,42)(32,38)(34,41)(36,48)(37,47)(39,44)(40,53)(43,51)(45,50)(46,52)(49,55)(54,56);(1,2,4,5)(3,7,8,6)(11,17)(12,20)(14,21)(15,19)(26,31)(28,39)(32,43)(34,42)(35,47)(36,49)(40,52)(50,56);(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27,30,38,29,37,41,51)(26,32,34,35,31,43,42,47)(28,36,40,50,39,49,52,56)(33,45,46,48,44,54,53,55);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,28,29,39)(26,33,31,44)(27,36,37,49)(30,40,41,52)(32,45,43,54)(34,46,42,53)(35,48,47,55)(38,50,51,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(28,39)(33,44)(36,49)(40,52)(45,54)(46,53)(48,55)(50,56)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gh,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104169);