Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1310 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)(4,8)(5,6)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,30,34)(27,35,36,50)(28,32,40,48)(29,42,43,33)(31,44,45,38)(37,51,39,54)(41,56,52,49)(46,55,47,53);(2,6)(3,4)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(27,36)(31,45)(35,50)(37,39)(38,44)(46,47)(51,54)(53,55);(1,2)(3,8)(4,7)(5,6)(25,27,29,38,30,36,43,44)(26,31,33,35,34,45,42,50)(28,39,41,55,40,37,52,53)(32,47,49,54,48,46,56,51);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,32)(27,37)(29,41)(30,40)(31,46)(33,49)(34,48)(35,51)(36,39)(38,53)(42,56)(43,52)(44,55)(45,47)(50,54);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(28,40)(32,48)(37,39)(41,52)(46,47)(49,56)(51,54)(53,55)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7199954);