Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 3145728 |
| 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 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,13,12)(11,14,16,15)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(9,10,13,12)(11,15,16,14)(18,22)(23,25)(24,28)(30,31)(35,39)(36,42)(43,47)(44,48);(9,11,13,16)(10,14,12,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fh,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2696293);