Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,27,33,28,34,26,32)(23,30,36,45,24,37,39,29)(25,31,42,48,43,49,41,47)(35,46,40,52,38,50,51,44);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,28,24)(22,29,34,45)(25,40,43,51)(26,36,27,39)(30,33,37,32)(31,46,49,50)(35,41,38,42)(44,47,52,48);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,28,23)(22,30,34,37)(25,38,43,35)(26,36,27,39)(29,33,45,32)(31,46,49,50)(40,42,51,41)(44,48,52,47);(8,16)(11,20)(13,15)(17,19)(21,25,26,41,28,43,27,42)(22,31,32,47,34,49,33,48)(23,35,39,51,24,38,36,40)(29,44,37,50,45,52,30,46);(6,12)(7,14)(11,20)(13,15)(21,26,28,27)(22,32,34,33)(23,36,24,39)(25,41,43,42)(29,30,45,37)(31,47,49,48)(35,40,38,51)(44,46,52,50)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949378);