Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25723 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1877 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,27,23,28,21,36)(19,29,31,42,32,41,38,24)(20,26,34,43,35,25,37,44)(30,46,48,39,33,45,47,40);(11,12)(14,15)(17,19)(18,24)(20,33)(21,31)(22,38)(23,32)(25,40)(26,39)(27,41)(28,42)(29,36)(30,35)(34,48)(37,47)(43,46)(44,45);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20,23,35)(18,25,28,26)(19,30,32,33)(21,37,22,34)(24,39,42,40)(27,44,36,43)(29,45,41,46)(31,48,38,47);(9,12,13,11)(10,15,16,14)(17,20)(18,26)(19,30)(21,37)(22,34)(23,35)(24,40)(25,28)(27,43)(29,46)(31,48)(32,33)(36,44)(38,47)(39,42)(41,45);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,21,23,22)(18,27,28,36)(19,31,32,38)(20,34,35,37)(24,41,42,29)(25,43,26,44)(30,47,33,48)(39,45,40,46)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396379);