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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,17,18,14)(12,20,22,15)(19,23,21,24)(25,26)(27,35)(28,33)(29,31)(30,43)(32,38)(34,42)(36,50)(37,49)(39,45)(40,54)(41,56)(44,53)(46,52)(47,55)(48,51);(1,3)(2,6)(4,7)(5,8)(9,11,13,18)(10,14,16,17)(12,21,22,19)(15,24,20,23)(26,31)(28,39)(32,44)(34,43)(35,49)(36,40)(41,51)(52,55);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,24)(21,22)(25,27,30,38,29,37,42,53)(26,32,34,35,31,44,43,49)(28,40,41,55,39,36,51,52)(33,47,48,54,45,46,56,50);(3,8)(6,7)(11,18)(14,17)(19,21)(23,24)(25,28)(26,33)(27,36)(29,39)(30,41)(31,45)(32,46)(34,48)(35,50)(37,40)(38,52)(42,51)(43,56)(44,47)(49,54)(53,55);(9,12,13,22)(10,15,16,20)(11,19,18,21)(14,23,17,24)(28,39)(33,45)(36,40)(41,51)(46,47)(48,56)(50,54)(52,55)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fgi,e^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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093386);