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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,15,13,16)(17,18,23,27)(19,28,31,39)(20,25,34,44)(21,26,22,37)(24,35,40,30)(29,45,32,48)(33,42,36,43)(38,46,41,47);(1,3,5,8)(2,6,4,7)(10,11)(12,13);(1,2,5,4)(3,6,8,7)(9,11,14,10)(12,15,13,16)(17,19,21,35,23,31,22,30)(18,24,37,39,27,40,26,28)(20,32,36,47,34,29,33,46)(25,41,42,45,44,38,43,48);(1,4,5,2)(3,6,8,7)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,38)(26,43)(27,44)(28,45)(30,47)(31,32)(35,46)(37,42)(39,48)(40,41);(9,13,14,12)(10,15,11,16)(17,21,23,22)(18,26,27,37)(19,30,31,35)(20,33,34,36)(24,39,40,28)(25,42,44,43)(29,46,32,47)(38,45,41,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=fgi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7094696);