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) = C25×C4 |
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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,7,8,6)(18,22)(19,25)(20,28)(30,32)(33,34)(35,38)(36,41)(37,42)(39,44)(40,45)(43,47)(46,48);(9,10,12,14)(11,13,15,16)(20,28)(24,29)(27,31)(30,32)(34,35)(37,45)(39,41)(43,46);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,34)(36,39,44,41)(37,40,45,42)(43,48,46,47);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,44)(34,39,35,41)(37,46,45,43)(40,48,42,47);(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=fgi,d^2=h,e^2=f,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, 4126248);