Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(23,32)(24,29)(25,30)(26,40)(27,31)(28,35)(33,43)(34,47)(36,44)(37,46)(38,41)(39,45)(42,50)(48,52)(49,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,35,27,33,25,38)(22,28,30,32,31,41,40,43)(24,34,36,50,37,48,39,49)(29,42,45,47,46,51,44,52);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(23,33)(28,41)(32,43)(34,48)(35,38)(42,51)(47,52)(49,50);(8,16)(11,20)(13,15)(17,19)(21,24,27,37)(22,29,31,46)(23,34,33,48)(25,39,26,36)(28,42,41,51)(30,45,40,44)(32,47,43,52)(35,50,38,49);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,30,31,40)(23,35,33,38)(24,36,37,39)(28,43,41,32)(29,44,46,45)(34,49,48,50)(42,47,51,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418095);