Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469687 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16660 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,11)(5,17)(7,20)(8,22)(9,18)(12,27)(13,29)(14,26)(16,19)(21,31)(25,30)(28,32);(1,2)(3,12)(4,8)(5,18)(6,10)(7,14)(9,17)(11,20)(13,28)(15,22)(16,24)(19,23)(21,30)(25,31)(26,27)(29,32);(1,3,5,11,6,14,17,26)(2,7,9,20,10,12,18,27)(4,13,16,29,15,30,19,25)(8,21,23,32,22,28,24,31);(1,4)(2,8)(3,13,14,30)(5,19)(6,15)(7,21,12,28)(9,24)(10,22)(11,25,26,29)(16,17)(18,23)(20,31,27,32)Polycyclic presentation:
2^9|c^2=f,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72286082, #72286134, #72286242 |
| maximal normal subgroup of | #72286082, #72286134, #72286242 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376883);