Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(15,19)(16,20)(21,22,26,27)(23,28,32,35)(24,25,34,29)(30,36,33,31);(5,7,9,12)(6,10,11,8)(13,14,17,18)(15,20,19,16)(22,27)(25,29)(28,35)(31,36);(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,28)(24,33)(25,31)(26,32)(27,35)(29,36)(30,34);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24,26,34)(22,29,27,25)(23,30,32,33)(28,31,35,36);(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,25,26,29)(22,24,27,34)(23,31,32,36)(28,33,35,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3044762);