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 | |
1330 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)(12,21,13,16)(19,23,22,24)(25,26,31,37)(27,38,42,33)(28,35,29,44)(30,36,32,47)(34,45,52,43)(39,54,40,49)(41,51,46,50)(48,56,53,55);(1,2)(3,7)(4,5)(6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,33)(28,39)(29,40)(30,46)(31,42)(32,41)(34,48)(35,49)(36,51)(37,38)(43,56)(44,54)(45,55)(47,50)(52,53);(2,5)(6,7)(25,28,31,29)(26,34,37,52)(27,39,42,40)(30,43,32,45)(33,48,38,53)(35,36,44,47)(41,55,46,56)(49,51,54,50);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,31,28)(26,35,37,44)(27,40,42,39)(30,43,32,45)(33,49,38,54)(34,36,52,47)(41,55,46,56)(48,51,53,50);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,31,32)(26,36,37,47)(27,41,42,46)(28,43,29,45)(33,50,38,51)(34,44,52,35)(39,55,40,56)(48,49,53,54)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6700832);