Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | |
1837 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)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,22,37)(20,30,33,47)(21,26,36,43)(25,28,40,32)(29,38,34,41)(31,46,35,48)(39,44,42,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,34,36,29)(25,30,40,47)(26,41,43,38)(31,45,35,44)(39,48,42,46);(10,14)(11,15)(17,20,19,28,23,33,22,32)(18,25,37,47,27,40,24,30)(21,35,29,45,36,31,34,44)(26,42,41,46,43,39,38,48);(2,6)(5,7)(17,21,23,36)(18,26,27,43)(19,29,22,34)(20,31,33,35)(24,38,37,41)(25,39,40,42)(28,44,32,45)(30,46,47,48);(3,8)(5,7)(11,15)(13,16)(21,36)(26,43)(29,34)(31,35)(38,41)(39,42)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=ghi,d^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=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669310);