Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23923 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1952 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,37)(24,28)(25,32)(30,45)(31,44)(33,40)(35,43)(36,42)(38,41)(39,46)(47,48);(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,30)(21,31)(22,32)(23,38)(24,33)(26,39)(27,40)(28,41)(29,37)(34,48)(35,46)(36,47)(42,44)(43,45);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20)(18,26)(19,30)(21,24)(22,36)(23,35)(25,39)(27,43)(28,34)(29,44)(31,33)(32,47)(37,42)(38,46)(40,45)(41,48);(11,15)(13,16)(17,21,24,20)(18,26,28,34)(19,31,33,30)(22,35,23,36)(25,39,41,48)(27,43,37,42)(29,44,40,45)(32,46,38,47);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,22,24,23)(18,27,28,37)(19,32,33,38)(20,35,21,36)(25,40,41,29)(26,42,34,43)(30,46,31,47)(39,44,48,45)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6383981);