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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 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)(21,22,26,32)(23,33,34,28)(24,38,41,30)(25,31,27,29)(35,44,39,47)(36,45,37,43)(40,48,42,46)(49,52,50,51);(2,3)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,28)(24,39)(25,36)(26,34)(27,37)(29,45)(30,47)(31,43)(32,33)(35,41)(38,44)(40,50)(42,49)(46,51)(48,52);(1,2,4,3)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,29)(23,34)(25,27)(28,43)(30,46)(31,32)(33,45)(35,39)(38,48)(40,42)(44,51)(47,52);(6,12)(10,13)(11,18)(17,19)(21,24,26,41)(22,30,32,38)(23,35,34,39)(25,42,27,40)(28,44,33,47)(29,46,31,48)(36,50,37,49)(43,51,45,52);(1,3,4,2)(7,15)(10,13)(14,20)(17,19)(21,25,26,27)(22,31,32,29)(23,36,34,37)(24,40,41,42)(28,43,33,45)(30,46,38,48)(35,49,39,50)(44,52,47,51)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=i,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, 6715005);