Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427377 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1274 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)(3,7)(4,15,17,8)(5,19)(6,10)(9,20)(11,24)(12,22,16,25)(13,23)(14,21)(18,26,30,29)(27,32,28,31);(1,3)(2,7)(4,16)(5,11)(6,13)(8,25)(9,21)(10,23)(12,17)(14,20)(15,22)(18,28)(19,24)(26,32)(27,30)(29,31);(3,11)(5,20)(7,21)(9,19)(12,27)(13,14)(16,28)(18,30)(22,31)(23,24)(25,32)(26,29);(1,4)(2,8)(3,12,13,16)(5,18)(6,17)(7,22,23,25)(9,26)(10,15)(11,27,14,28)(19,29)(20,30)(21,31,24,32)Polycyclic presentation:
2^9|a^2=e,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318217, #72318430, #72318916, … |
| maximal normal subgroup of | #72318217, #72318430, #72318916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334573);