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 | #23871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,23,24)(20,31,33,44)(21,26,35,42)(25,37,40,29)(30,43,36,38)(32,47,34,48)(39,46,41,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,23)(18,24,27,28)(20,29,33,37)(21,30,35,36)(25,31,40,44)(26,38,42,43)(32,45,34,46)(39,47,41,48);(11,15)(13,16)(17,20,22,33)(18,25,27,40)(19,29,23,37)(21,34,35,32)(24,31,28,44)(26,41,42,39)(30,46,36,45)(38,48,43,47);(11,15)(13,16)(17,21,22,35)(18,26,27,42)(19,30,23,36)(20,32,33,34)(24,38,28,43)(25,39,40,41)(29,45,37,46)(31,47,44,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(21,35)(26,42)(30,36)(32,34)(38,43)(39,41)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6379263);