Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364101 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13959 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 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)(3,12,16,22)(4,17,19,25)(5,9)(6,11)(7,15,23,13)(8,18,24,31)(10,20)(14,28)(21,30)(26,27)(29,32);(1,3,5,15,6,16,20,13)(2,7,10,22,11,23,9,12)(4,14,18,29,19,30,31,27)(8,21,25,32,24,28,17,26);(3,13)(5,20)(7,12)(9,10)(14,27)(15,16)(17,25)(18,31)(21,26)(22,23)(28,32)(29,30);(1,4,6,19)(2,8)(3,14,16,30)(5,18,20,31)(7,21)(9,17)(10,25)(11,24)(12,26)(13,27,15,29)(22,32)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216077, #72216181, … |
| maximal normal subgroup of | #72215972, #72216077, #72216181, … |
| maximal quotient of | #72226688, #72226689, C2×#364101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271297);