Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31)(33,34,38,41)(35,36,42,39)(37,45,46,40)(43,47,48,44);(9,10,13,12)(11,15,16,14)(18,22)(19,26)(20,28)(31,32)(35,42)(36,39)(43,48)(44,47);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,35)(34,39)(36,41)(37,43)(38,42)(40,47)(44,45)(46,48);(1,3,5,7)(2,6,4,8)(9,10,13,12)(11,14,16,15)(17,19,21,26)(18,23,22,25)(20,29,28,27)(24,32,30,31)(33,36,38,39)(34,35,41,42)(37,44,46,47)(40,43,45,48);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,28)(18,24,22,30)(19,27,26,29)(23,31,25,32)(33,37,38,46)(34,40,41,45)(35,43,42,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3238729);