Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16690 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
931 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)(4,16,18,9)(5,8)(6,11)(7,14)(10,19)(13,21,26,25)(15,22)(17,24,29,23)(20,28)(27,30,32,31);(1,3)(2,7,11,12)(4,13)(5,15)(6,14)(8,20,10,22)(9,21,16,25)(17,27)(18,26)(19,28)(23,30,24,31)(29,32);(2,8)(5,19)(7,20)(9,23)(10,11)(12,22)(13,26)(15,28)(16,24)(17,29)(21,31)(25,30);(1,4)(2,9)(3,13)(5,17)(6,18)(7,21)(8,23)(10,24)(11,16)(12,25)(14,26)(15,27)(19,29)(20,30)(22,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72286082, #72286134, #72286246 |
| maximal normal subgroup of | #72286082, #72286134, #72286246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377785);