Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(10,13)(14,16)(18,23)(20,25)(21,30)(26,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2)(3,6)(4,7)(5,8)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=gi,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1757494);