Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34636 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1808 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)(10,14)(15,19)(16,18)(17,20)(21,22,25,30,26,29,37,44)(23,27,32,40,33,39,47,51)(24,28,36,43,35,42,45,49)(31,38,46,50,34,41,48,52);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,25,32,26,33,37,47)(22,27,30,40,29,39,44,51)(24,34,36,48,35,31,45,46)(28,41,43,52,42,38,49,50);(2,4)(7,14)(10,13)(15,20)(17,19)(21,24)(22,28)(23,31)(25,36)(26,35)(27,38)(29,42)(30,43)(32,46)(33,34)(37,45)(39,41)(40,50)(44,49)(47,48)(51,52);(6,11)(10,13)(12,18)(17,19)(24,35)(28,42)(31,34)(36,45)(38,41)(43,49)(46,48)(50,52);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(22,29)(27,39)(28,42)(30,44)(38,41)(40,51)(43,49)(50,52)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cfi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283120);