Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116392 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
809 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)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18,21,24,22,26,20,25)(19,23,27,31,28,32,29,30);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,31)(25,30)(26,32);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=ei,g^c=ghi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312200, #72312211, #72312749, … |
| maximal normal subgroup of | #72312200, #72312211, #72312749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23590);