Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
1755 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,28)(23,31,34,27)(24,36,38,29)(26,30,40,45)(32,44,35,42)(33,43,51,49)(37,46,50,52)(39,48,41,47);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,34)(22,27,28,31)(24,37,38,50)(26,32,40,35)(29,46,36,52)(30,42,45,44)(33,39,51,41)(43,47,49,48);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,32)(26,40)(27,42)(30,45)(31,44)(33,50)(34,35)(37,51)(39,41)(43,52)(46,49)(47,48);(6,12)(10,13)(11,18)(17,19)(22,28)(24,38)(27,31)(30,45)(33,51)(37,50)(39,41)(42,44);(7,15)(10,13)(14,20)(17,19)(21,24,25,38)(22,29,28,36)(23,33,34,51)(26,41,40,39)(27,43,31,49)(30,48,45,47)(32,50,35,37)(42,52,44,46)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576965);