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 | #38032 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1535 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)(19,28)(20,24)(21,25)(22,27)(23,31)(26,29)(30,42)(32,37)(33,39)(34,40)(35,41)(36,46)(38,43)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(20,33)(24,39)(30,44)(34,35)(36,47)(40,41)(42,48)(45,46);(2,6)(3,8)(10,13)(14,16)(17,19,29,43,22,32,21,31)(18,23,25,37,27,38,26,28)(20,30,34,45,33,44,35,46)(24,36,41,48,39,47,40,42);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,30)(21,35)(22,33)(23,36)(25,41)(26,40)(27,39)(28,42)(29,34)(31,46)(32,44)(37,48)(38,47)(43,45);(10,13)(14,16)(17,21,22,29)(18,25,27,26)(19,31,32,43)(20,34,33,35)(23,37,38,28)(24,40,39,41)(30,45,44,46)(36,42,47,48)Polycyclic presentation:
2^9|c^2=fgi,d^2=fh,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693477);