Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 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)(3,6)(4,7)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,41)(38,51)(39,45)(40,48)(42,46)(43,50)(44,49)(47,53)(52,56)(54,55);(1,3,6,2)(4,7,5,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(26,32)(28,42)(33,45)(35,36)(37,48)(38,52)(43,44)(53,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,39,30,40,31,41)(26,33,35,48,32,45,36,37)(28,38,44,54,42,52,43,53)(34,47,50,56,46,55,49,51);(1,4,6,5)(2,7,3,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,42)(26,34,32,46)(27,38,40,52)(29,44,31,43)(33,47,45,55)(35,50,36,49)(37,51,48,56)(39,54,41,53);(1,5,6,4)(2,7,3,8)(11,18)(14,22)(17,19)(21,23)(25,29,30,31)(26,35,32,36)(27,39,40,41)(28,43,42,44)(33,48,45,37)(34,49,46,50)(38,53,52,54)(47,51,55,56)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7178732);