Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(9,10,14,11)(12,17,20,15)(13,18,21,16)(19,22,24,23)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36)(41,42,46,44)(43,49,51,47)(45,52,54,48)(50,55,53,56);(10,11)(13,21)(15,17)(19,24)(27,32)(28,31)(35,40)(36,39)(41,43,46,51)(42,47,44,49)(45,53,54,50)(48,56,52,55);(1,2,4,6)(3,5,7,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36)(41,43)(42,47)(44,49)(45,50)(46,51)(48,55)(52,56)(53,54);(1,3,4,7)(2,5,6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35)(41,44,46,42)(43,47,51,49)(45,52,54,48)(50,55,53,56);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)(41,45)(42,48)(43,50)(44,52)(46,54)(47,55)(49,56)(51,53)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fhi,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4430770);