Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39335 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1970 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,31,38)(20,25,32,40)(21,26,22,36)(24,34,39,30)(29,43,44,48)(33,41,35,42)(37,45,47,46);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(20,32)(25,40)(29,44)(33,35)(37,47)(41,42)(43,48)(45,46);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,19,21,34,23,31,22,30)(18,24,36,38,27,39,26,28)(20,29,35,45,32,44,33,46)(25,37,41,48,40,47,42,43);(2,6)(5,7)(10,14)(13,15)(17,20,23,32)(18,25,27,40)(19,29,31,44)(21,35,22,33)(24,37,39,47)(26,42,36,41)(28,43,38,48)(30,46,34,45);(3,8)(5,7)(17,21,23,22)(18,26,27,36)(19,30,31,34)(20,33,32,35)(24,38,39,28)(25,41,40,42)(29,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=i,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=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815308);