Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1634 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)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,30,26,29,36,42)(23,27,33,39,32,38,45,49)(24,28,34,41,35,40,46,50)(31,37,44,48,43,47,51,52);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,27)(24,31)(25,33)(26,32)(28,37)(29,38)(30,39)(34,44)(35,43)(36,45)(40,47)(41,48)(42,49)(46,51)(50,52);(2,4)(21,24,25,34,26,35,36,46)(22,28,30,41,29,40,42,50)(23,31,33,44,32,43,45,51)(27,37,39,48,38,47,49,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,29)(23,32)(28,40)(30,42)(31,43)(33,45)(41,50)(44,51);(6,12)(10,13)(11,20)(18,19)(23,32)(27,38)(31,43)(33,45)(37,47)(39,49)(44,51)(48,52)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6299940);