Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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)(21,22,27,28,26,32,25,31)(23,33,36,43,34,29,37,41)(24,30,40,42,39,46,38,45)(35,47,49,52,48,44,50,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(23,34)(24,38)(25,27)(29,43)(31,32)(33,41)(35,49)(39,40)(42,45)(44,47)(48,50);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23)(22,29)(24,35)(25,37)(26,34)(27,36)(28,41)(30,44)(31,43)(32,33)(38,50)(39,48)(40,49)(42,51)(45,52)(46,47);(7,14)(10,13)(15,20)(18,19)(21,24,27,40,26,39,25,38)(22,30,28,42,32,46,31,45)(23,35,36,49,34,48,37,50)(29,44,41,51,33,47,43,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,26,27)(22,28,32,31)(23,36,34,37)(24,38,39,40)(29,43,33,41)(30,42,46,45)(35,49,48,50)(44,52,47,51)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fi,e^2=h,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=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090827);