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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,12)(15,16)(18,22)(19,25)(24,29)(26,31)(33,34,38,42)(35,40,36,39)(37,41,45,48)(43,47,44,46);(2,5)(6,8)(10,14)(15,16)(19,25)(23,28)(26,31)(30,32)(33,35,38,36)(34,39,42,40)(37,44,45,43)(41,47,48,46);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,12,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gi,e^2=gh,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, 4431335);