Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #38035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1530 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)(17,18,22,27)(19,23,30,26)(20,31,33,36)(21,25,35,42)(24,38,40,28)(29,37,34,41)(32,44,46,47)(39,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,26)(20,28,33,38)(21,34,35,29)(24,36,40,31)(25,41,42,37)(32,45,46,43)(39,44,48,47);(2,6)(3,8)(10,13)(14,16)(17,20,30,38,22,33,19,28)(18,24,23,36,27,40,26,31)(21,32,34,45,35,46,29,43)(25,39,37,47,42,48,41,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,29)(20,32)(22,35)(23,37)(24,39)(26,41)(27,42)(28,43)(30,34)(31,44)(33,46)(36,47)(38,45)(40,48);(10,13)(14,16)(21,35)(25,42)(29,34)(32,46)(37,41)(39,48)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693500);