Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207830 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1507 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,33)(30,45)(31,38)(32,41)(34,39)(35,48)(36,44)(37,43)(40,46)(42,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(20,34)(25,38)(27,28)(29,41)(30,35)(36,37)(46,47);(11,15)(13,16)(17,19,21,31,22,32,23,33)(18,25,27,41,24,38,28,29)(20,35,37,46,34,30,36,47)(26,42,44,45,39,40,43,48);(3,8)(5,7)(10,14)(11,15)(17,20,22,34)(18,26,24,39)(19,30,32,35)(21,37,23,36)(25,40,38,42)(27,44,28,43)(29,45,41,48)(31,47,33,46);(2,6)(5,7)(11,15)(13,16)(17,21,22,23)(18,27,24,28)(19,31,32,33)(20,36,34,37)(25,41,38,29)(26,43,39,44)(30,46,35,47)(40,45,42,48)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7115345);