Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21503 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,15,5,14)(3,11,6,23,18,13,17,9)(4,19,21,29,22,32,24,28)(10,20,26,16,27,30,25,31);(1,3,5,17,8,18,7,6)(2,9,12,11,15,23,14,13)(4,16,21,30,22,31,24,20)(10,19,25,28,27,32,26,29);(1,4)(2,10)(3,16)(5,21)(6,20)(7,24)(8,22)(9,19)(11,28)(12,25)(13,29)(14,26)(15,27)(17,30)(18,31)(23,32);(1,3,7,6,8,18,5,17)(2,11,12,23,15,13,14,9)(4,20,21,16,22,30,24,31)(10,19,26,29,27,32,25,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,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 | #72214039, #72214066, #72214124, … |
| maximal normal subgroup of | #72214039, #72214066, #72214124, … |
| maximal quotient of | #72216353, #72216356, #72216367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403050);