Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | |
1203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,32)(29,33)(30,35)(31,39)(34,37)(38,50)(40,45)(41,47)(42,49)(43,48)(44,53)(46,51)(52,55)(54,56);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,37,51,30,40,29,39)(26,31,33,45,35,46,34,36)(28,38,41,52,43,54,42,53)(32,44,49,56,48,55,47,50);(1,4,5,3)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,43)(26,32,35,48)(27,38,40,54)(29,42,37,41)(31,44,46,55)(33,49,34,47)(36,50,45,56)(39,53,51,52);(10,16)(11,18)(15,24)(17,19)(25,29,30,37)(26,33,35,34)(27,39,40,51)(28,41,43,42)(31,45,46,36)(32,47,48,49)(38,52,54,53)(44,50,55,56)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fgi,d^2=fi,e^2=i,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=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6951902);