Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1893 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,28)(23,31,34,44)(24,29,35,42)(25,37,38,30)(27,39,40,33)(32,45,46,52)(36,49,48,43)(41,50,51,47);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,33,26,34,38,39)(22,27,30,31,28,40,37,44)(24,32,36,47,35,46,48,50)(29,41,43,45,42,51,49,52);(1,2,3,4)(7,13)(10,17)(14,19)(18,20)(22,28)(27,40)(29,42)(30,37)(31,44)(41,51)(43,49)(45,52);(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,32)(25,36)(26,35)(27,41)(28,42)(30,43)(31,45)(33,47)(34,46)(37,49)(38,48)(39,50)(40,51)(44,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,35)(29,42)(32,46)(36,48)(41,51)(43,49)(45,52)(47,50)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404839);