Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,7)(8,11,12,10)(13,14,17,15)(16,19,20,18)(21,22,26,25)(23,28,31,27)(24,29,34,33)(30,36,32,35);(2,4)(6,7)(8,12)(21,23,26,31)(22,27,25,28)(24,30,34,32)(29,35,33,36);(1,2,3,4)(6,7)(10,11)(13,15,17,14)(16,19,20,18)(21,23)(22,28)(24,32)(25,27)(26,31)(29,35)(30,34)(33,36);(5,7,9,6)(8,11,12,10)(13,16,17,20)(14,18,15,19)(21,24)(22,29)(23,30)(25,33)(26,34)(27,35)(28,36)(31,32);(5,8)(6,10)(7,11)(9,12)(21,25,26,22)(23,27,31,28)(24,33,34,29)(30,35,32,36)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fghi,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1758876);