Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 | #37445 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1757 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,32)(21,26)(22,37)(23,28)(25,34)(27,30)(29,42)(31,40)(33,47)(35,44)(36,41)(38,43)(39,46)(45,48);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,29)(21,22)(23,31)(25,39)(26,37)(27,41)(28,40)(30,36)(32,42)(33,38)(34,46)(35,45)(43,47)(44,48);(11,16)(13,15)(17,20,23,35)(18,25,28,43)(19,29,31,45)(21,33,36,46)(22,38,30,34)(24,39,40,47)(26,42,41,48)(27,44,37,32);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,21)(18,26)(19,30)(20,33)(22,31)(23,36)(24,27)(25,42)(28,41)(29,34)(32,47)(35,46)(37,40)(38,45)(39,44)(43,48);(2,5)(6,7)(19,31)(21,36)(24,40)(26,41)(29,45)(33,46)(39,47)(42,48)Polycyclic presentation:
2^9|c^2=i,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6551791);