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 | |
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:
(3,7)(6,8)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,39,42)(35,37,36,40)(38,41,47,48)(43,45,44,46);(10,13)(14,16)(18,22)(23,25)(24,28)(31,32)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,46,48,45);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,36,39,35)(34,40,42,37)(38,43,47,44)(41,45,48,46);(1,3,5,7)(2,6,4,8)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,37,39,40)(34,35,42,36)(38,46,47,45)(41,44,48,43);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,38,39,47)(34,41,42,48)(35,43,36,44)(37,45,40,46)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=hi,d^2=gi,e^2=ghi,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, 3721004);