Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1188 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)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,33)(28,43)(29,35)(30,36)(31,46)(32,37)(34,44)(38,52)(39,48)(40,49)(41,55)(42,50)(45,51)(47,53)(54,56);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,32,42)(26,33,37,50)(28,38,29,39)(30,40,31,41)(34,47,51,56)(35,48,43,52)(36,49,46,55)(44,53,45,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,32,29)(26,34,37,51)(27,38,42,39)(30,44,31,45)(33,47,50,56)(35,46,43,36)(40,53,41,54)(48,55,52,49);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,29,32,28)(26,35,37,43)(27,39,42,38)(30,44,31,45)(33,48,50,52)(34,46,51,36)(40,53,41,54)(47,55,56,49);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,30,32,31)(26,36,37,46)(27,40,42,41)(28,44,29,45)(33,49,50,55)(34,35,51,43)(38,53,39,54)(47,48,56,52)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=i,d^2=fi,e^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=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6947502);