Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,27,31)(23,28,33,42)(24,36,37,50)(25,30,26,40)(29,38,44,39)(32,45,48,52)(34,43,35,49)(41,46,51,47);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,28)(24,32)(25,34)(26,35)(27,33)(29,41)(30,43)(31,42)(36,45)(37,48)(38,46)(39,47)(40,49)(44,51)(50,52);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,39,27,37,25,38)(22,29,30,36,31,44,40,50)(23,32,35,47,33,48,34,46)(28,41,43,45,42,51,49,52);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(23,33)(24,37)(28,42)(29,44)(34,35)(36,50)(38,39)(43,49);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,25,27,26)(22,30,31,40)(23,34,33,35)(24,38,37,39)(28,43,42,49)(29,36,44,50)(32,46,48,47)(41,45,51,52)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594547);