Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,11)(9,10)(13,14,17,19)(15,16,20,18)(21,22)(23,32)(24,28)(25,34)(26,30)(27,31)(29,35)(33,36);(6,9)(7,8)(13,14)(15,18)(16,20)(17,19)(21,23,27,24)(22,28,31,32)(25,30,35,33)(26,34,36,29);(5,7,10,8)(6,11,9,12)(13,15,17,20)(14,18,19,16)(21,24,27,23)(22,28,31,32)(25,33,35,30)(26,34,36,29);(1,2,3,4)(5,8,10,7)(6,11,9,12)(13,16)(14,15)(17,18)(19,20)(21,25,27,35)(22,29,31,34)(23,33,24,30)(26,28,36,32);(5,9,10,6)(7,11,8,12)(13,15)(14,16)(17,20)(18,19)(21,26,27,36)(22,30,31,33)(23,34,24,29)(25,28,35,32)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=f,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990256);