Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1538 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,29,20,24)(21,35,38,48)(22,39,30,27)(25,33,41,32)(26,43,45,37)(31,42,34,44)(36,46,47,40);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,36)(22,32)(25,27)(26,44)(28,29)(30,33)(31,37)(34,43)(35,40)(38,47)(39,41)(42,45)(46,48);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,32,23,19,30,33)(18,25,27,24,28,41,39,29)(21,34,37,47,38,31,43,36)(26,42,35,40,45,44,48,46);(1,4,5,3)(2,6,8,7)(9,11,12,16)(10,14,13,15)(17,21,30,43,23,38,22,37)(18,26,39,48,28,45,27,35)(19,31,32,47,20,34,33,36)(24,40,25,42,29,46,41,44);(2,8)(6,7)(17,22,23,30)(18,27,28,39)(19,32,20,33)(21,37,38,43)(24,25,29,41)(26,35,45,48)(31,47,34,36)(40,42,46,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fhi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503392);