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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,29,43)(20,24,32,40)(21,34,35,25)(23,36,38,30)(28,42,31,46)(33,48,44,41)(37,47,39,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,29,35,36)(18,23,25,27,26,38,34,43)(20,31,33,47,32,28,44,45)(24,39,41,46,40,37,48,42);(2,6)(7,8)(11,16)(13,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,42)(29,31)(30,45)(34,48)(35,44)(36,47)(38,39)(43,46);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(19,29)(20,32)(23,38)(24,40)(27,43)(30,36)(33,44)(41,48)Polycyclic presentation:
2^9|a^2=i,c^2=fg,e^2=f,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, 6934587);