Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1388 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)(12,16)(14,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,34,38,41)(35,36,42,39)(37,45,46,40)(43,47,48,44);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(18,22)(19,26)(24,28)(27,29)(35,42)(36,39)(43,48)(44,47);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,28)(21,22)(24,30)(25,26)(27,32)(29,31)(33,35)(34,39)(36,41)(37,43)(38,42)(40,47)(44,45)(46,48);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,29)(21,26)(22,25)(24,32)(27,30)(28,31)(33,36,38,39)(34,35,41,42)(37,44,46,47)(40,43,45,48);(17,20,21,30)(18,24,22,28)(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=hi,c^2=fgi,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3476253);