Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #34609 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12,13,7)(5,11)(6,8)(10,14,16,15)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30);(1,3,9,2,6,11,5,8)(4,10,14,7,13,16,15,12)(17,19,21,30,22,28,20,27)(18,23,24,31,25,32,29,26);(1,4)(2,7)(3,10)(5,15)(6,13)(8,12)(9,14)(11,16);(4,13)(7,12)(10,16)(14,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312041, #72312052 |
| maximal normal subgroup of | #72312041, #72312052 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104540);