Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,21,26,22,27,36,44)(19,28,30,39,31,43,35,23)(20,25,34,42,33,24,40,41)(29,46,47,37,32,45,48,38);(9,10)(11,15)(12,13)(14,16)(17,19,22,31)(18,23,27,39)(20,32,33,29)(21,35,36,30)(24,38,25,37)(26,43,44,28)(34,47,40,48)(41,45,42,46);(2,6)(7,8)(10,13)(14,15)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,27)(26,41)(28,45)(30,47)(31,32)(35,48)(36,40)(38,39)(42,44)(43,46);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,29)(21,34)(22,33)(23,38)(24,27)(26,42)(28,46)(30,47)(31,32)(35,48)(36,40)(37,39)(41,44)(43,45);(1,4,5,3)(2,8,6,7)(20,33)(24,25)(29,32)(34,40)(37,38)(41,42)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376988);