Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1860 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)(23,33)(24,37)(25,30)(26,42)(27,31)(28,35)(29,40)(32,43)(34,47)(36,46)(38,44)(39,51)(41,49)(45,50)(48,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,35,27,36,43,50)(22,28,32,33,31,45,42,46)(24,38,40,52,39,25,34,41)(29,30,37,48,47,44,51,49);(7,13)(10,17)(14,19)(18,20)(21,24,26,40,27,39,43,34)(22,29,32,37,31,47,42,51)(23,25,35,41,36,38,50,52)(28,44,33,49,45,30,46,48);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,34)(24,35)(26,41)(27,38)(28,37)(29,46)(31,44)(32,48)(33,47)(36,40)(39,50)(42,49)(43,52)(45,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,31)(24,39)(25,38)(28,45)(32,42)(33,46)(34,40)(41,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404939);