Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1641 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,28,34)(23,30,37,43)(24,29,36,44)(25,31,41,48)(26,32,27,33)(35,45,50,51)(38,42,49,52)(39,46,40,47);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,36,28,37,26,24)(22,29,32,43,34,44,33,30)(25,35,39,49,41,50,40,38)(31,42,47,51,48,52,46,45);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24,26,37,28,36,27,23)(22,30,33,44,34,43,32,29)(25,38,40,50,41,49,39,35)(31,45,46,52,48,51,47,42);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,25)(22,31)(23,35)(24,38)(26,40)(27,39)(28,41)(29,42)(30,45)(32,47)(33,46)(34,48)(36,49)(37,50)(43,51)(44,52);(6,12)(10,17)(11,20)(18,19)(21,26,28,27)(22,32,34,33)(23,24,37,36)(25,39,41,40)(29,43,44,30)(31,46,48,47)(35,49,50,38)(42,45,52,51)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6389472);