Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1372 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,15,16,13)(19,25)(20,27)(22,28)(23,30)(33,34)(35,39)(36,44)(37,38)(40,45)(41,48)(42,43)(46,47);(9,11,14,16)(10,15,12,13)(19,25)(22,28)(26,31)(29,32)(34,37)(36,45)(39,42)(41,46);(2,3)(6,7)(10,12)(11,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,38,43)(34,39,37,42)(36,46,45,41)(40,48,44,47);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,16,13)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36)(34,40)(35,41)(37,44)(38,45)(39,47)(42,48)(43,46);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,16,12,11)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,37,38,34)(35,42,43,39)(36,44,45,40)(41,48,46,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926881);