Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 | |
1583 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)(17,18,22,27)(19,29,32,24)(20,26,35,42)(21,28,23,25)(30,43,45,38)(31,37,33,39)(34,40,36,41)(44,48,46,47);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,27,29)(20,30,35,45)(21,31,23,33)(25,37,28,39)(26,38,42,43)(34,44,36,46)(40,47,41,48);(9,11)(10,13)(12,16)(14,15)(18,25)(21,23)(24,37)(26,40)(27,28)(29,39)(31,33)(34,36)(38,47)(41,42)(43,48)(44,46);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,20,22,35)(18,26,27,42)(19,30,32,45)(21,36,23,34)(24,38,29,43)(25,40,28,41)(31,46,33,44)(37,47,39,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,25,27,28)(19,31,32,33)(20,34,35,36)(24,37,29,39)(26,41,42,40)(30,44,45,46)(38,48,43,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,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=dfhi,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7215520);