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 | |
1661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,16)(11,17)(15,20)(18,19)(21,22,27,33,28,34,26,32)(23,35,25,31,37,47,41,29)(24,30,39,45,40,46,42,48)(36,49,38,44,51,52,50,43);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,29)(24,36)(25,26)(27,41)(28,37)(30,43)(31,34)(32,35)(33,47)(38,42)(39,50)(40,51)(44,46)(45,52)(48,49);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,28,40)(22,30,34,46)(23,36,37,51)(25,38,41,50)(26,42,27,39)(29,43,31,44)(32,48,33,45)(35,49,47,52);(7,14)(10,13)(15,20)(18,19)(21,25,26,23,28,41,27,37)(22,31,32,35,34,29,33,47)(24,38,42,36,40,50,39,51)(30,44,48,49,46,43,45,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,32,34,33)(23,25,37,41)(24,39,40,42)(29,35,31,47)(30,45,46,48)(36,50,51,38)(43,52,44,49)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=i,d^2=fi,e^2=hi,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091140);