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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 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)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22,27,32)(23,33,36,47)(24,29,38,45)(25,31,26,30)(28,40,43,35)(34,49,37,52)(39,48,41,46)(42,50,44,51);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,40,27,36,26,35)(22,28,30,47,32,43,31,33)(24,37,41,51,38,34,39,50)(29,44,48,49,45,42,46,52);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,38)(22,29,32,45)(23,34,36,37)(25,41,26,39)(28,42,43,44)(30,48,31,46)(33,49,47,52)(35,51,40,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,35)(24,38)(25,26)(28,33)(29,45)(30,31)(34,50)(36,40)(37,51)(42,52)(43,47)(44,49);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,35,36,40)(24,39,38,41)(28,33,43,47)(29,46,45,48)(34,50,37,51)(42,52,44,49)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032778);