Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,25,30,26,28,38,42)(23,31,33,39,34,44,37,27)(24,29,36,43,35,41,48,51)(32,45,47,50,46,52,49,40);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,34)(22,27,28,39)(24,32,35,46)(25,37,38,33)(29,40,41,50)(30,44,42,31)(36,49,48,47)(43,52,51,45);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(22,28)(27,39)(29,41)(30,42)(31,44)(40,50)(43,51)(45,52);(7,13)(10,17)(14,19)(18,20)(21,24,26,35)(22,29,28,41)(23,32,34,46)(25,36,38,48)(27,40,39,50)(30,43,42,51)(31,45,44,52)(33,47,37,49);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,35)(29,41)(32,46)(36,48)(40,50)(43,51)(45,52)(47,49)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405851);