Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,28)(24,30)(25,32)(26,31)(27,33)(29,38)(34,42)(35,44)(36,43)(37,45)(39,47)(40,46)(41,48)(49,52)(50,51);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,37)(22,28,33,45)(24,34,39,49)(25,35,26,36)(29,41,46,51)(30,42,47,52)(31,43,32,44)(38,48,40,50);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,39)(22,29,33,46)(23,34,37,49)(25,38,26,40)(28,41,45,51)(30,32,47,31)(35,48,36,50)(42,44,52,43);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,30)(23,34)(25,40)(26,38)(27,39)(28,42)(29,31)(32,46)(33,47)(35,50)(36,48)(37,49)(41,43)(44,51)(45,52);(6,12)(7,14)(11,20)(13,15)(21,25,27,26)(22,31,33,32)(23,35,37,36)(24,38,39,40)(28,43,45,44)(29,30,46,47)(34,48,49,50)(41,42,51,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6941980);