Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1229 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)(3,6)(4,9)(5,20,23,14)(7,11)(8,12)(10,22,28,24)(13,17)(15,19)(16,30,31,27)(18,26)(21,29,32,25);(1,3,7,12,8,11,6,2)(4,15,18,26,19,9,17,13)(5,14,22,28,23,20,24,10)(16,29,21,30,31,25,32,27);(1,4)(2,9)(3,13)(5,21,23,32)(6,17)(7,18)(8,19)(10,27,28,30)(11,26)(12,15)(14,29,20,25)(16,22,31,24);(1,5,6,24,8,23,7,22)(2,10,11,20,12,28,3,14)(4,16,17,32,19,31,18,21)(9,25,26,30,15,29,13,27)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72286066, #72286186, #72286246, … |
| maximal normal subgroup of | #72286066, #72286186, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367324);