Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1241 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)(11,19,21,15)(12,16,13,18)(20,24,22,23)(25,26,30,35)(27,31,38,47)(28,33,40,51)(29,42,44,34)(32,43,50,41)(36,52,37,46)(39,56,55,48)(45,54,49,53);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,31)(28,37)(29,39)(30,38)(32,49)(33,46)(34,48)(35,47)(36,40)(41,54)(42,56)(43,53)(44,55)(45,50)(51,52);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,36)(29,43)(30,40)(31,45)(33,42)(34,51)(35,50)(37,38)(39,54)(41,44)(46,48)(47,49)(52,56)(53,55);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,33)(27,37)(29,41)(30,40)(31,46)(32,42)(34,50)(35,51)(36,38)(39,54)(43,44)(45,48)(47,52)(49,56)(53,55);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,38)(28,40)(31,47)(32,50)(33,51)(39,55)(41,43)(48,56)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6684530);