Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25727 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1877 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,23,37,22,28,21,27)(19,29,31,24,33,40,32,41)(20,26,34,43,36,25,35,42)(30,45,48,39,47,44,46,38);(11,12)(14,15)(17,19)(18,24)(20,30)(21,31)(22,33)(23,32)(25,38)(26,39)(27,40)(28,41)(29,37)(34,46)(35,48)(36,47)(42,44)(43,45);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20,22,36)(18,25,28,26)(19,30,33,47)(21,35,23,34)(24,38,41,39)(27,43,37,42)(29,44,40,45)(31,48,32,46);(9,12,13,11)(10,15,16,14)(17,20)(18,26)(19,30)(21,35)(22,36)(23,34)(24,39)(25,28)(27,42)(29,45)(31,48)(32,46)(33,47)(37,43)(38,41)(40,44);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,21,22,23)(18,27,28,37)(19,31,33,32)(20,34,36,35)(24,40,41,29)(25,42,26,43)(30,46,47,48)(38,44,39,45)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6396372);