Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14727 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,8)(3,13,14,24,17,23,16,9)(4,10,19,21,20,26,18,25)(15,29,30,32,31,27,28,22);(2,8)(3,14)(5,6)(10,21)(11,12)(13,23)(15,28)(16,17)(18,19)(25,26)(27,29)(30,31);(1,3,5,16,7,17,6,14)(2,9,11,23,12,24,8,13)(4,15,18,30,20,31,19,28)(10,22,25,29,26,32,21,27);(1,4)(2,10)(3,15)(5,18)(6,19)(7,20)(8,21)(9,22)(11,25)(12,26)(13,27)(14,28)(16,30)(17,31)(23,29)(24,32)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72317787, #72317872, #72318482, … |
| maximal normal subgroup of | #72317787, #72317872, #72318482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302987);