Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #23871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1631 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)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,26)(21,27)(22,37)(23,28)(24,29)(30,39)(31,41)(32,42)(33,40)(34,43)(35,45)(36,46)(38,44)(47,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,20)(18,25,29,26)(21,34,36,47)(22,31,23,33)(27,43,46,48)(28,40,37,41)(30,38,32,35)(39,44,42,45);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,24,19)(18,26,29,25)(21,32,36,30)(22,31,23,33)(27,42,46,39)(28,40,37,41)(34,35,47,38)(43,45,48,44);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21)(18,27)(19,30)(20,32)(22,35)(23,38)(24,36)(25,39)(26,42)(28,44)(29,46)(31,47)(33,34)(37,45)(40,43)(41,48);(2,6)(7,8)(17,22,24,23)(18,28,29,37)(19,31,20,33)(21,35,36,38)(25,40,26,41)(27,44,46,45)(30,47,32,34)(39,43,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433056);