Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | #25761 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1654 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)(3,6)(4,7)(5,8)(17,18,22,24)(19,29,32,43)(20,26,33,39)(21,28,23,27)(25,36,38,31)(30,45,34,48)(35,44,37,42)(40,46,41,47);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(25,38)(27,28)(29,43)(30,34)(35,37)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,36,22,32,23,31)(18,25,27,43,24,38,28,29)(20,34,37,47,33,30,35,46)(26,41,44,45,39,40,42,48);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,20,22,33)(18,26,24,39)(19,30,32,34)(21,37,23,35)(25,40,38,41)(27,44,28,42)(29,45,43,48)(31,47,36,46);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,24,28)(19,31,32,36)(20,35,33,37)(25,29,38,43)(26,42,39,44)(30,46,34,47)(40,48,41,45)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=f,e^2=f,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=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7244300);