Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1750 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,27,32)(23,33,36,46)(24,29,38,44)(25,31,26,30)(28,40,42,35)(34,47,49,51)(37,45,39,43)(41,48,52,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23)(22,28)(24,34)(25,35)(26,40)(27,36)(29,41)(30,33)(31,46)(32,42)(37,48)(38,49)(39,50)(43,51)(44,52)(45,47);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24)(22,29)(23,34)(25,39)(26,37)(27,38)(28,41)(30,45)(31,43)(32,44)(33,47)(35,50)(36,49)(40,48)(42,52)(46,51);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,32,31)(23,35,36,40)(24,37,38,39)(28,33,42,46)(29,43,44,45)(34,48,49,50)(41,51,52,47);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,38)(29,44)(34,49)(37,39)(41,52)(43,45)(47,51)(48,50)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405659);