Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | #39382 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22,25,29)(23,31,33,27)(24,35,37,51)(26,39,36,30)(28,42,43,38)(32,46,48,50)(34,41,47,45)(40,49,44,52);(2,3)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,32)(25,33)(26,34)(28,40)(29,31)(30,41)(35,50)(36,47)(37,48)(38,49)(39,45)(42,52)(43,44)(46,51);(1,2,4,3)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,36,42,25,37,26,38)(22,28,39,51,29,43,30,35)(23,32,47,52,33,48,34,49)(27,40,45,46,31,44,41,50);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(22,29)(23,33)(28,43)(30,39)(32,48)(34,47)(35,51)(49,52);(1,3,4,2)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,33)(27,31)(32,48)(34,47)(40,44)(41,45)(46,50)(49,52)Polycyclic presentation:
2^9|a^2=f,c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724618);