Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(21,22,25,29,26,30,37,44)(23,27,33,40,31,38,45,49)(24,28,36,43,35,42,48,52)(32,39,46,50,34,41,47,51);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,34)(25,33)(26,31)(28,41)(29,40)(30,38)(32,35)(36,47)(37,45)(39,42)(43,51)(44,49)(46,48)(50,52);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,31)(27,38)(32,34)(33,45)(39,41)(40,49)(46,47)(50,51);(7,15)(10,13)(14,20)(17,19)(21,24)(22,28)(23,32)(25,36)(26,35)(27,39)(29,43)(30,42)(31,34)(33,46)(37,48)(38,41)(40,50)(44,52)(45,47)(49,51);(6,12)(10,13)(11,18)(17,19)(23,31)(24,35)(27,38)(28,42)(33,45)(36,48)(40,49)(43,52)Polycyclic presentation:
2^9|a^2=h,c^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293300);