Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(18,22)(20,27)(23,28)(26,31)(33,34,38,39)(35,40,44,47)(36,37,41,46)(42,45,48,43);(9,10,13,12)(11,15,16,14)(19,25)(23,28)(26,31)(30,32)(34,39)(36,41)(40,47)(42,48);(9,11,13,16)(10,14,12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,40)(36,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,16,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,37)(35,42)(38,41)(39,46)(40,45)(43,47)(44,48);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,41)(35,43)(36,39)(38,46)(40,42)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gh,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3566698);