Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(6,7)(11,15)(14,16)(19,25)(20,27)(23,28)(24,29)(33,34,38,39)(35,42,41,37)(36,44,46,40)(43,47,48,45);(9,10,13,12)(11,14,15,16)(18,22)(23,28)(24,29)(30,32)(34,39)(35,41)(40,44)(43,48);(1,2,5,3)(4,6,8,7)(9,11,13,15)(10,14,12,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,41)(34,37,39,42)(36,43,46,48)(40,47,44,45);(1,3,5,2)(4,7,8,6)(9,10,13,12)(11,14,15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,46)(34,40,39,44)(35,43,41,48)(37,47,42,45);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,41)(35,39)(36,45)(38,42)(40,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=fghi,c^2=fgh,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4337067);