Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #23871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,4,8)(9,10,13,12)(11,14,15,16)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,29)(30,40)(32,44)(33,42)(34,37)(35,41)(36,39)(38,43)(45,48)(46,47);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,22,23)(18,24,27,28)(20,29,33,36)(21,34,35,30)(25,31,39,42)(26,40,41,37)(32,45,46,43)(38,47,48,44);(10,12)(11,15)(17,20,22,33)(18,25,27,39)(19,29,23,36)(21,32,35,46)(24,31,28,42)(26,38,41,48)(30,43,34,45)(37,44,40,47);(1,3,5,4)(2,6,7,8)(9,12,13,10)(11,14,15,16)(17,21,22,35)(18,26,27,41)(19,30,23,34)(20,32,33,46)(24,37,28,40)(25,38,39,48)(29,43,36,45)(31,44,42,47);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(18,27)(21,35)(24,28)(25,39)(30,34)(31,42)(32,46)(43,45)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7316809);