Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 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)(5,6,9,13)(7,11,14,10)(8,16,18,12)(15,19,17,20)(21,22,27,31)(23,28,26,40)(24,34,36,47)(25,30,38,44)(29,32,43,37)(33,41,39,50)(35,46,49,52)(42,45,51,48);(1,2)(3,4)(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,23,27,26)(22,28,31,40)(24,32,36,37)(25,33,38,39)(29,34,43,47)(30,41,44,50)(35,45,49,48)(42,46,51,52);(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,24,26,37,27,36,23,32)(22,29,28,34,31,43,40,47)(25,35,39,48,38,49,33,45)(30,42,41,46,44,51,50,52);(2,4)(7,14)(10,11)(15,17)(19,20)(21,25,27,38)(22,30,31,44)(23,33,26,39)(24,35,36,49)(28,41,40,50)(29,42,43,51)(32,45,37,48)(34,46,47,52);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(24,36)(25,38)(29,43)(30,44)(32,37)(33,39)(34,47)(41,50)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7100473);