Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,31)(21,25)(22,35)(23,29)(24,34)(26,32)(28,41)(30,37)(33,46)(36,48)(38,47)(39,44)(40,42)(43,45);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,45)(32,42)(33,43)(34,44)(35,40)(38,39)(41,46)(47,48);(1,3)(2,7)(4,6)(5,8)(11,16)(13,15)(17,20,32,47,21,33,22,34)(18,24,35,46,25,38,26,31)(19,28,42,48,29,43,30,44)(23,36,40,41,27,39,37,45);(2,6)(3,8)(10,14)(13,15)(18,25)(19,29)(24,38)(26,35)(28,43)(30,42)(31,46)(44,48);(1,4,5,7)(2,3,6,8)(19,29)(23,27)(28,43)(30,42)(36,39)(37,40)(41,45)(44,48)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6877473);