Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1580 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,29,24,30,22,28)(19,31,34,42,35,44,39,25)(20,36,37,43,21,27,38,26)(32,40,48,41,33,46,47,45);(2,5)(6,8)(10,14)(11,12)(17,19)(18,25)(20,32)(21,33)(22,34)(23,39)(24,35)(26,40)(27,41)(28,31)(29,44)(30,42)(36,45)(37,47)(38,48)(43,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,24,21)(18,26,30,43)(19,32,35,33)(22,37,23,38)(25,40,42,46)(27,28,36,29)(31,45,44,41)(34,47,39,48);(9,12,13,11)(10,16,14,15)(17,21,24,20)(18,27,30,36)(19,33,35,32)(22,37,23,38)(25,41,42,45)(26,28,43,29)(31,46,44,40)(34,47,39,48);(10,14)(15,16)(17,22,24,23)(18,28,30,29)(19,34,35,39)(20,37,21,38)(25,31,42,44)(26,36,43,27)(32,47,33,48)(40,45,46,41)Polycyclic presentation:
2^9|a^2=g,c^2=fg,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391771);