Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,31,35)(27,32,39,49)(28,33,43,52)(29,34,44,30)(36,47,54,55)(37,38,48,45)(40,50,53,56)(41,42,51,46);(1,3,4,7)(2,5,6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,32)(28,40)(29,37)(30,45)(31,39)(33,50)(34,38)(35,49)(36,42)(41,55)(43,53)(44,48)(46,54)(47,51)(52,56);(2,6)(5,8)(25,28,31,43)(26,33,35,52)(27,36,39,54)(29,41,44,51)(30,46,34,42)(32,47,49,55)(37,50,48,56)(38,53,45,40);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29)(26,34)(27,37)(28,41)(30,35)(31,44)(32,38)(33,42)(36,50)(39,48)(40,55)(43,51)(45,49)(46,52)(47,53)(54,56);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,38)(29,44)(30,34)(32,48)(36,53)(37,49)(39,45)(40,54)(41,51)(42,46)(47,56)(50,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589300);