Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 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)(3,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,29)(24,30)(25,31)(26,41)(27,33)(28,34)(32,42)(35,43)(36,44)(37,45)(38,46)(39,48)(40,49)(47,51)(50,52);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,28,36)(22,29,34,44)(24,37,40,51)(25,26,27,42)(30,45,49,47)(31,41,33,32)(35,38,50,39)(43,46,52,48);(2,4)(21,24,27,39,28,40,25,38)(22,30,33,48,34,49,31,46)(23,35,26,37,36,50,42,51)(29,43,41,45,44,52,32,47);(6,11)(10,13)(12,19)(18,20)(21,25,28,27)(22,31,34,33)(23,26,36,42)(24,38,40,39)(29,41,44,32)(30,46,49,48)(35,37,50,51)(43,45,52,47);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,26,27,36,28,42,25,23)(22,32,33,29,34,41,31,44)(24,37,39,50,40,51,38,35)(30,47,48,43,49,45,46,52)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716965);