Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 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)(21,22)(23,32)(24,29)(25,30)(26,41)(27,31)(28,35)(33,49)(34,44)(36,43)(37,52)(38,46)(39,47)(40,48)(42,50)(45,51);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,35,27,36,25,34)(22,28,30,32,31,44,41,43)(24,37,38,51,40,33,39,50)(29,45,47,52,48,42,46,49);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,27,40)(22,29,31,48)(23,33,36,37)(25,39,26,38)(28,42,44,45)(30,47,41,46)(32,49,43,52)(34,51,35,50);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,38,40,39)(28,32,44,43)(29,46,48,47)(33,50,37,51)(42,52,45,49);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,31,41)(23,35,36,34)(24,39,40,38)(28,43,44,32)(29,47,48,46)(33,50,37,51)(42,52,45,49)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6454385);