Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #118350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,8)(6,10)(7,13)(11,15)(14,16)(17,18)(19,22)(20,27,29,23)(21,24)(25,32,28,30)(26,31);(1,3,4,10,5,11,13,16)(2,6,7,9,8,14,12,15)(17,19)(18,22)(20,28,29,25)(21,26)(23,30,27,32)(24,31);(6,14)(9,15)(17,20)(18,23)(19,25)(21,29)(22,30)(24,27)(26,28)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312642, #72312646, #72312650, … |
| maximal normal subgroup of | #72312642, #72312646, #72312650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25546);