Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,28,22,27,38,26)(19,29,33,46,32,23,31,41)(20,25,37,43,36,45,35,24)(30,44,47,39,34,40,48,42);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,23,27,29)(20,34,36,30)(21,33,38,31)(24,42,43,39)(25,44,45,40)(26,46,28,41)(35,47,37,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,24,27,43)(19,30,32,34)(21,35,38,37)(23,39,29,42)(25,28,45,26)(31,47,33,48)(40,41,44,46);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,37)(22,36)(23,40)(24,26)(27,45)(28,43)(29,44)(31,48)(32,34)(33,47)(35,38)(39,46)(41,42);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,38)(18,26,27,28)(19,31,32,33)(20,35,36,37)(23,41,29,46)(24,25,43,45)(30,47,34,48)(39,44,42,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6918095);