Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1377 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,36)(28,41)(29,35)(30,32)(31,46)(33,40)(34,42)(37,53)(38,54)(39,47)(43,52)(44,51)(45,50)(48,56)(49,55);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(26,32)(27,37)(28,42)(35,46)(39,40)(44,45)(48,49)(52,54);(2,6)(3,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,31,40,30,37,29,39)(26,33,35,36,32,47,46,53)(28,43,44,56,42,38,45,55)(34,49,51,54,41,48,50,52);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,34)(27,38)(29,45)(30,42)(31,44)(32,41)(33,48)(35,51)(36,52)(37,43)(39,56)(40,55)(46,50)(47,49)(53,54);(3,7)(5,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,35,32,46)(27,39,37,40)(28,44,42,45)(33,36,47,53)(34,50,41,51)(38,55,43,56)(48,54,49,52)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fgh,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6673662);