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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1335 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)(4,6)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,34,31,35,43,49)(27,36,38,45,39,48,29,32)(28,33,42,47,41,46,53,55)(37,50,52,54,51,56,40,44);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,39)(26,32,35,45)(28,37,41,51)(29,43,38,30)(33,44,46,54)(34,48,49,36)(40,53,52,42)(47,56,55,50);(2,6)(5,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,33)(27,37)(29,40)(30,42)(31,41)(32,44)(34,47)(35,46)(36,50)(38,52)(39,51)(43,53)(45,54)(48,56)(49,55);(3,8)(5,7)(12,20)(15,24)(17,19)(21,23)(25,29,30,27,31,38,43,39)(26,32,34,36,35,45,49,48)(28,40,42,37,41,52,53,51)(33,44,47,50,46,54,55,56);(10,16)(11,18)(15,24)(17,19)(28,41)(33,46)(37,51)(40,52)(42,53)(44,54)(47,55)(50,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=h,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7018055);