Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324436 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,33)(30,43)(31,37)(32,40)(34,38)(35,42)(36,41)(39,45)(44,48)(46,47);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(18,24)(20,34)(25,37)(27,28)(29,40)(30,44)(35,36)(45,46);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,19,21,31,22,32,23,33)(18,25,27,40,24,37,28,29)(20,30,36,46,34,44,35,45)(26,39,42,48,38,47,41,43);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,38)(25,39)(27,42)(28,41)(29,43)(31,46)(32,44)(33,45)(37,47)(40,48);(2,6)(5,7)(10,11)(14,15)(17,21,22,23)(18,27,24,28)(19,31,32,33)(20,35,34,36)(25,40,37,29)(26,41,38,42)(30,45,44,46)(39,43,47,48)Polycyclic presentation:
2^9|b^2=fi,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7237819);