Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,31)(24,28)(25,37)(26,29)(27,34)(30,38)(32,43)(33,44)(35,42)(36,49)(39,46)(40,47)(41,48)(45,51)(50,52);(2,3)(7,13)(8,16)(10,17)(11,15)(23,32)(27,39)(31,43)(33,45)(34,46)(40,50)(44,51)(47,52);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23,25,34,26,32,38,46)(22,27,30,31,29,39,37,43)(24,33,36,47,35,45,48,52)(28,40,41,44,42,50,49,51);(1,3,4,2)(7,13)(10,17)(14,19)(18,20)(21,24,26,35)(22,28,29,42)(23,33,32,45)(25,36,38,48)(27,40,39,50)(30,41,37,49)(31,44,43,51)(34,47,46,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,29)(24,35)(27,39)(30,37)(31,43)(33,45)(36,48)(47,52)Polycyclic presentation:
2^9|a^2=fhi,c^2=ghi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103918);