Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1787 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,32)(23,28,33,44)(24,30,38,48)(25,31,26,29)(34,42,36,46)(35,43,39,41)(37,45,40,47)(49,51,50,52);(2,3)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,27,33)(22,28,32,44)(24,36,38,34)(25,39,26,35)(29,43,31,41)(30,46,48,42)(37,49,40,50)(45,51,47,52);(1,2,4,3)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,29)(23,33)(25,26)(28,41)(30,45)(31,32)(34,36)(37,40)(42,51)(43,44)(46,52)(47,48);(6,12)(10,13)(11,18)(17,19)(21,24)(22,30)(23,34)(25,40)(26,37)(27,38)(28,42)(29,45)(31,47)(32,48)(33,36)(35,50)(39,49)(41,51)(43,52)(44,46);(1,3,4,2)(7,15)(10,13)(14,20)(17,19)(21,25,27,26)(22,31,32,29)(23,35,33,39)(24,37,38,40)(28,43,44,41)(30,45,48,47)(34,49,36,50)(42,51,46,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715015);