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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,32)(28,40)(29,38)(31,45)(33,46)(34,47)(35,42)(36,48)(39,44)(41,43);(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)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(11,15)(13,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,37,46)(21,34,36,43,35,31,29,44)(26,41,38,47,42,39,48,45);(2,6)(5,7)(17,21)(18,26)(19,29)(20,31)(22,36)(23,35)(24,38)(25,39)(27,42)(28,43)(30,45)(32,44)(33,34)(37,48)(40,41)(46,47);(3,8)(5,7)(11,15)(13,16)(21,35)(26,42)(29,36)(31,34)(38,48)(39,41)(43,44)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6657222);