Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,15,11,8,20,14)(7,12,19,26,18,16,29,25)(9,13,23,28,22,21,33,27)(17,24,31,35,32,30,36,34);(1,2,3,4)(5,7)(6,12)(8,16)(9,17)(10,19)(11,18)(13,24)(14,25)(15,26)(20,29)(21,30)(22,32)(23,31)(27,34)(28,35)(33,36);(5,8)(6,11)(7,16)(9,21)(10,14)(12,18)(13,22)(15,20)(17,30)(19,25)(23,27)(24,32)(26,29)(28,33)(31,34)(35,36);(5,9,10,23,11,22,20,33)(6,13,15,28,8,21,14,27)(7,17,19,31,18,32,29,36)(12,24,26,35,16,30,25,34);(7,18)(9,22)(12,16)(13,21)(19,29)(23,33)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,g^c=gi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72320787 |
| maximal normal subgroup of | #72320787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280037);