Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437969 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,13,10,14,22,20)(7,11,16,24,17,25,21,28)(8,12,19,27,18,26,31,32)(15,23,30,35,29,34,33,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,20)(10,14)(13,22)(15,23)(16,28)(17,25)(18,26)(19,32)(21,24)(27,31)(29,34)(30,36)(33,35);(2,4)(5,7,10,17)(6,11,14,25)(8,15,18,29)(9,21,22,16)(12,23,26,34)(13,28,20,24)(19,33,31,30)(27,36,32,35);(2,4)(5,8)(6,12)(7,15)(9,19)(10,18)(11,23)(13,27)(14,26)(16,30)(17,29)(20,32)(21,33)(22,31)(24,35)(25,34)(28,36);(8,18)(12,26)(15,29)(19,31)(23,34)(27,32)(30,33)(35,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312597, #72312619, #72312662, … |
| maximal normal subgroup of | #72312597, #72312619, #72312662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345167);