Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500060 |
| 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×#3219 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,26,31,27,32,25,30)(23,28,36,44,34,42,35,43)(24,37,38,45,39,46,40,29)(33,47,49,51,48,52,50,41);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,28)(24,33)(25,35)(26,36)(27,34)(29,41)(30,43)(31,44)(32,42)(37,47)(38,49)(39,48)(40,50)(45,51)(46,52);(6,11)(7,13)(8,15)(16,20)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,41)(30,37)(31,46)(32,45)(34,48)(35,49)(36,50)(42,51)(43,47)(44,52);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(23,34)(28,42)(33,48)(35,36)(41,51)(43,44)(47,52)(49,50);(7,13)(10,17)(14,19)(16,20)(21,25,27,26)(22,30,32,31)(23,35,34,36)(24,38,39,40)(28,43,42,44)(29,37,45,46)(33,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=gh,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6432136);