Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #37659 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1793 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,28,36)(20,25,31,40)(21,24,33,27)(29,35,41,47)(30,37,42,34)(32,38,44,39)(43,46,48,45);(2,6)(4,7)(10,13)(11,15)(17,19,22,28)(18,23,26,36)(20,29,31,41)(21,30,33,42)(24,37,27,34)(25,35,40,47)(32,43,44,48)(38,46,39,45);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(20,31)(21,33)(23,34)(25,39)(26,27)(35,46)(36,37)(38,40)(43,48)(45,47);(2,6)(3,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(28,41)(30,43)(33,44)(34,45)(36,47)(37,46)(42,48);(1,4,5,7)(2,8,6,3)(18,26)(19,28)(20,31)(24,27)(30,42)(32,44)(35,47)(45,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6871390);