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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1734 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)(21,22,25,30,26,29,39,45)(23,31,34,40,32,46,38,27)(24,28,37,43,36,44,51,52)(33,47,48,42,35,49,50,41);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,32)(22,27,29,40)(24,35,36,33)(25,38,39,34)(28,42,44,41)(30,46,45,31)(37,48,51,50)(43,47,52,49);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(23,32)(27,40)(31,46)(33,35)(34,38)(41,42)(47,49)(48,50);(7,13)(10,17)(14,19)(18,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,41)(29,44)(30,43)(31,47)(32,35)(34,48)(38,50)(39,51)(40,42)(45,52)(46,49);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,29)(24,36)(27,40)(30,45)(31,46)(33,35)(37,51)(48,50)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,e^2=hi,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, 6405855);