Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72316063, #72316085, #72316119, … |
| maximal normal subgroup of | #72316063, #72316085, #72316119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108588);