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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1290 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)(9,10,13,11)(12,15,16,14)(17,18)(19,28)(20,33)(21,37)(22,26)(23,27)(24,32)(25,35)(29,39)(30,40)(31,41)(34,42)(36,43)(38,44)(45,48)(46,47);(2,6)(7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,24)(20,34)(21,31)(22,32)(23,30)(25,42)(26,28)(27,41)(29,35)(33,39)(36,45)(37,40)(38,46)(43,47)(44,48);(1,3,5,4)(2,7,6,8)(9,12,13,16)(10,14,11,15)(17,20)(18,25)(19,29)(21,38)(22,35)(23,36)(24,39)(26,33)(27,44)(28,42)(30,46)(31,45)(32,34)(37,43)(40,48)(41,47);(1,4,5,3)(2,8,6,7)(10,11)(14,15)(18,26)(19,30)(20,35)(21,23)(24,40)(28,41)(29,45)(31,32)(34,46)(39,47)(42,48)(43,44);(2,6)(7,8)(17,21,22,23)(18,27,26,37)(19,31,32,30)(20,36,35,38)(24,41,28,40)(25,43,33,44)(29,46,34,45)(39,48,42,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7452408);