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 | C2×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,34)(24,31)(25,30)(26,32)(27,43)(28,33)(29,38)(35,45)(36,44)(37,46)(39,50)(40,51)(41,49)(42,48)(47,52);(7,13)(8,16)(10,17)(11,15)(21,23)(22,29)(24,40)(25,35)(26,38)(27,37)(28,39)(30,47)(31,44)(32,50)(33,46)(34,43)(36,42)(41,52)(45,48)(49,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24)(22,30)(23,35)(25,28)(26,42)(27,41)(29,44)(31,33)(32,49)(34,51)(36,37)(38,52)(39,40)(43,48)(45,50)(46,47);(7,13)(10,17)(14,19)(18,20)(21,25,28,24)(22,31,33,30)(23,36,39,52)(26,42,27,41)(29,45,46,51)(32,49,43,48)(34,47,50,44)(35,38,40,37);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,32,33,43)(23,37,39,38)(24,41,25,42)(29,34,46,50)(30,48,31,49)(35,52,40,36)(44,45,47,51)Polycyclic presentation:
2^9|a^2=hi,c^2=gi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7081021);