Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,29,34,30,33,41,48)(27,31,37,44,36,43,50,54)(28,32,40,46,39,47,52,56)(35,42,49,53,38,45,51,55);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27)(26,31)(28,38)(29,37)(30,36)(32,45)(33,43)(34,44)(35,39)(40,51)(41,50)(42,47)(46,55)(48,54)(49,52)(53,56);(2,5)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,32)(27,35)(29,40)(30,39)(31,42)(33,47)(34,46)(36,38)(37,49)(41,52)(43,45)(44,53)(48,56)(50,51)(54,55);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(26,33)(28,39)(31,43)(34,48)(35,38)(40,52)(44,54)(49,51);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,36)(31,43)(35,38)(37,50)(42,45)(44,54)(49,51)(53,55)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6316895);