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 | #30708 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1658 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)(17,18,23,27)(19,28,29,39)(20,25,32,40)(21,26,22,36)(24,34,37,31)(30,43,44,48)(33,41,35,42)(38,45,47,46);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(19,29)(20,32)(24,37)(25,40)(28,39)(31,34)(33,35)(41,42);(9,11)(10,13)(12,16)(14,15)(17,19,21,34,23,29,22,31)(18,24,36,39,27,37,26,28)(20,30,35,45,32,44,33,46)(25,38,41,48,40,47,42,43);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,20,23,32)(18,25,27,40)(19,30,29,44)(21,35,22,33)(24,38,37,47)(26,42,36,41)(28,43,39,48)(31,46,34,45);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,31,29,34)(20,33,32,35)(24,39,37,28)(25,41,40,42)(30,45,44,46)(38,43,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=fi,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=ehi,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, 7244423);