Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500060 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3562 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,27,42,26,31,25,30)(23,32,41,28,34,48,35,44)(24,36,39,29,38,47,40,46)(33,43,50,49,37,45,51,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,28)(24,37)(25,41)(26,34)(27,35)(29,45)(30,48)(31,44)(32,42)(33,38)(36,52)(39,50)(40,51)(43,46)(47,49);(6,11)(7,13)(8,15)(16,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,40)(28,43)(30,47)(31,46)(32,49)(34,37)(35,50)(36,42)(41,51)(44,45)(48,52);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(23,34)(24,38)(28,44)(29,46)(32,48)(35,41)(36,47)(39,40);(7,13)(10,17)(14,19)(16,20)(21,25,26,27)(22,30,31,42)(23,35,34,41)(24,39,38,40)(28,32,44,48)(29,47,46,36)(33,50,37,51)(43,49,45,52)Polycyclic presentation:
2^9|a^2=gh,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6431952);