Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,17,31)(8,11)(9,22)(10,25)(12,18,26,34)(13,24)(14,21)(16,33,19,30)(20,32)(23,28,36,29)(27,35);(1,2,3,4)(6,11)(12,26)(13,27)(14,25)(15,31)(22,32)(28,29)(30,33);(5,7)(6,12)(8,17)(9,23)(10,16)(11,26)(13,30)(14,28)(15,32)(18,35)(19,21)(20,36)(22,31)(24,34)(25,29)(27,33);(7,16,17,19)(9,20)(12,28,26,29)(13,27)(15,33,31,30)(18,36,34,23)(22,32)(24,35)Polycyclic presentation:
2^9|a^2=g,b^2=ei,d^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250341, #72250863, #72265093, … |
| maximal normal subgroup of | #72250341, #72250863, #72265093, … |
| maximal quotient of | #72287608, #72287703, #72287758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403035);