Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(19,24)(20,25)(21,26)(22,28)(23,27)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(45,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,27,39)(20,32,34,45)(21,30,22,35)(25,40,42,47)(26,38,28,43)(29,36,46,33)(37,44,48,41);(1,4,5,6)(2,3,7,8)(10,14)(13,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,41)(27,42)(28,44)(30,45)(31,46)(32,35)(38,47)(39,48)(40,43);(2,7)(3,8)(11,16)(13,15)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,43)(25,41,42,44)(29,45,46,32)(37,47,48,40);(18,27)(24,39)(25,42)(26,28)(37,48)(38,43)(40,47)(41,44)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414621);