Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1603 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,33)(21,27)(22,28)(23,29)(24,32)(25,36)(30,39)(31,44)(34,41)(35,45)(37,40)(38,43)(42,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,28,26)(20,34,36,46)(21,31,23,37)(25,41,33,48)(27,40,29,44)(30,38,47,35)(39,45,42,43);(10,14)(13,15)(17,20,22,36)(18,25,28,33)(19,30,32,47)(21,35,23,38)(24,39,26,42)(27,43,29,45)(31,46,37,34)(40,48,44,41);(2,7)(3,8)(11,16)(13,15)(17,19)(18,26)(20,30)(21,37)(22,32)(23,31)(24,28)(25,42)(27,40)(29,44)(33,39)(34,35)(36,47)(38,46)(41,45)(43,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,28,29)(19,31,32,37)(20,35,36,38)(24,40,26,44)(25,43,33,45)(30,46,47,34)(39,48,42,41)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419060);