Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632068 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,14,18)(11,19,21,15)(12,17,13,16)(20,23,22,24)(25,26,31,37)(27,39,42,33)(28,35,29,45)(30,38,32,36)(34,47,52,44)(40,49,46,53)(41,50,43,51)(48,55,54,56);(1,2,4,5)(3,6,7,8)(9,11,14,21)(10,15,18,19)(12,22,13,20)(16,24,17,23)(25,27)(26,33)(28,40)(29,46)(30,41)(31,42)(32,43)(34,48)(35,53)(36,50)(37,39)(38,51)(44,55)(45,49)(47,56)(52,54);(2,5)(6,8)(25,28,31,29)(26,34,37,52)(27,40,42,46)(30,44,32,47)(33,48,39,54)(35,38,45,36)(41,55,43,56)(49,50,53,51);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,29,31,28)(26,35,37,45)(27,40,42,46)(30,44,32,47)(33,49,39,53)(34,38,52,36)(41,56,43,55)(48,50,54,51);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,31,32)(26,36,37,38)(27,41,42,43)(28,44,29,47)(33,50,39,51)(34,35,52,45)(40,55,46,56)(48,53,54,49)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=i,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6565724);