Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1411 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,24,39,23,28,22,29)(19,30,38,25,32,44,21,27)(20,33,40,43,36,26,37,47)(31,46,48,42,34,45,35,41);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19)(18,25)(20,34)(21,24)(22,38)(23,32)(26,42)(27,28)(29,44)(30,39)(31,36)(33,41)(35,37)(40,48)(43,45)(46,47);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,26)(19,31)(21,35)(22,37)(23,36)(24,40)(25,41)(27,42)(28,33)(29,43)(30,45)(32,34)(38,48)(39,47)(44,46);(10,13)(14,16)(17,21,24,19,23,38,22,32)(18,27,39,30,28,25,29,44)(20,35,40,31,36,48,37,34)(26,42,47,45,33,41,43,46);(9,11,12,15)(10,14,13,16)(18,28)(19,32)(20,36)(21,38)(29,39)(37,40)(41,42)(45,46)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091776);