Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470867 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2018 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)(23,33)(24,38)(25,30)(26,32)(27,31)(28,37)(29,40)(34,43)(35,45)(36,44)(39,46)(41,48)(42,47)(49,51)(50,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,25,37,27,36,26,35)(22,28,32,33,31,45,30,44)(24,39,41,51,42,34,40,50)(29,46,47,52,48,43,38,49);(6,11)(7,13)(12,15)(14,19)(21,24,26,40,27,42,25,41)(22,29,30,38,31,48,32,47)(23,34,35,51,36,39,37,50)(28,43,44,52,45,46,33,49);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,32)(23,35,36,37)(24,40,42,41)(28,33,45,44)(29,47,48,38)(34,50,39,51)(43,52,46,49);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(23,36)(29,48)(30,32)(34,39)(35,37)(38,47)(50,51)Polycyclic presentation:
2^9|b^2=f,c^2=ghi,d^2=i,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=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406055);