Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470860 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,33)(24,38)(25,30)(26,31)(27,32)(28,37)(29,40)(34,42)(35,44)(36,43)(39,46)(41,45)(47,49)(48,52)(50,51);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,26,36,27,35,25,37)(22,28,30,44,32,43,31,33)(24,34,40,49,41,48,39,50)(29,42,38,51,46,52,45,47);(7,13)(10,17)(14,15)(18,19)(21,24,25,39,27,41,26,40)(22,29,31,45,32,46,30,38)(23,34,37,50,35,48,36,49)(28,42,33,47,43,52,44,51);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,35)(28,43)(33,44)(34,48)(36,37)(42,52)(47,51)(49,50);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,33,43,44)(29,38,46,45)(34,49,48,50)(42,47,52,51)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=h,e^2=i,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^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391859);