Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460970 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2759 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16390 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,14,7)(4,8,17,26)(5,9)(6,10)(11,20)(13,24)(15,23,29,22)(16,21)(18,25,19,27)(28,32)(30,31);(1,3)(2,7)(4,16,17,13)(5,15)(6,14)(8,24,26,21)(9,22)(10,12)(11,23)(18,30,19,28)(20,29)(25,32,27,31);(1,4)(2,8,10,26)(3,13)(5,19)(6,17)(7,21,12,24)(9,27,11,25)(14,16)(15,30)(18,20)(22,32,23,31)(28,29);(2,9,10,11)(7,22,12,23)(8,25,26,27)(21,31,24,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286106, #72286218, #72286246 |
| maximal normal subgroup of | #72286106, #72286218, #72286246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368590);