Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,4,8,6)(12,16)(14,15)(18,22)(20,27)(23,28)(26,31)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(1,3,5,8)(2,6,7,4)(9,10,13,11)(12,14,16,15)(18,22)(19,25)(24,29)(26,31)(35,42)(38,41)(43,48)(45,47);(2,7)(4,6)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,3)(2,4)(5,8)(6,7)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35)(34,38)(36,43)(37,42)(39,45)(40,41)(44,48)(46,47);(1,4)(2,3)(5,6)(7,8)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,44)(34,39,40,46)(35,43,42,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=fgi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637400);