Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,12)(11,16,13,15)(19,25)(20,27)(22,28)(23,30)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,4)(6,7)(9,10,14,12)(11,15,13,16)(19,25)(22,28)(26,31)(29,32)(34,35)(36,43)(40,42)(44,48);(2,4)(6,7)(9,11,14,13)(10,15,12,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,15,13,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,15,12,16)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=ghi,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6159855);