Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,33)(29,32)(30,35)(34,41)(36,43)(37,42)(38,45)(39,48)(40,46)(44,51)(47,52)(49,55)(50,53)(54,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,30,38)(26,31,35,45)(28,36,39,49)(29,37,41,51)(32,42,34,44)(33,43,48,55)(40,50,52,56)(46,53,47,54);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(28,39)(29,41)(31,42)(33,47)(34,35)(36,49)(37,51)(43,54)(44,45)(46,48)(53,55);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,36)(29,40)(30,39)(31,43)(32,46)(34,47)(35,48)(37,50)(38,49)(41,52)(42,53)(44,54)(45,55)(51,56);(10,16)(14,17)(15,24)(22,23)(28,39)(33,48)(36,49)(40,52)(43,55)(46,47)(50,56)(53,54)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032143);