Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,32)(29,43)(30,33)(31,39)(34,37)(36,49)(38,50)(40,54)(41,48)(42,56)(44,51)(45,53)(46,55)(47,52);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,36)(31,44)(35,49)(38,40)(39,51)(45,46)(50,54)(53,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,37,51,30,36,29,39)(26,31,43,49,33,44,34,35)(28,40,52,53,41,38,42,55)(32,46,56,50,48,45,47,54);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,41)(26,32,33,48)(27,38,36,40)(29,42,37,52)(31,45,44,46)(34,47,43,56)(35,50,49,54)(39,53,51,55);(10,16)(11,18)(15,24)(17,19)(26,33)(28,41)(31,44)(34,43)(35,49)(38,40)(42,52)(53,55)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949997);