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 | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,4,17)(3,12,16,23,13,26,14,8)(7,9,19,24,20,21,18,25)(15,27,30,31,28,32,29,22);(1,3,7,15)(2,8,21,31)(4,14,18,29)(5,16,19,30)(6,13,20,28)(9,22,11,23)(10,12,25,32)(17,26,24,27);(2,9)(4,18)(5,19)(8,22)(11,21)(14,29)(16,30)(23,31);(3,13)(8,23)(12,26)(14,16)(15,28)(22,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=egi,e^2=h,g^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 | #72239753, #72241259, #72241261, … |
| maximal normal subgroup of | #72239753, #72241259, #72241261, … |
| maximal quotient of | #72257809 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108662);