Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1302 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)(3,6,7,8)(11,15)(13,16)(18,22)(20,28)(23,25)(26,32)(33,34,38,39)(35,36,41,44)(37,40,47,48)(42,43,45,46);(3,7)(6,8)(9,10,12,14)(11,13,15,16)(20,28)(24,29)(26,32)(30,31)(34,39)(35,41)(40,48)(42,45);(1,3,5,7)(2,6,4,8)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,36)(37,45)(38,41)(39,44)(40,46)(42,47)(43,48);(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,35)(37,46)(38,44)(39,41)(40,45)(42,48)(43,47);(1,4,5,2)(3,8,7,6)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,47)(34,40,39,48)(35,42,41,45)(36,43,44,46)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fgh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4087007);