Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,14,15,16)(17,18)(19,24)(20,26)(21,25)(22,23)(27,31)(28,32)(29,30)(34,38)(35,41)(36,44)(46,48);(11,15)(14,16)(18,23)(19,26)(21,29)(28,31)(33,34,37,38)(35,42,41,39)(36,45,44,40)(43,46,47,48);(1,2,5,4)(3,6,8,7)(9,11,13,15)(10,14,12,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,3,5,8)(2,6,4,7)(9,10,13,12)(11,14,15,16)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,16,15,14)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=fghi,d^2=fghi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1820406);