Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359617 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,5,21)(3,13,18,29,16,8,17,25)(4,9,19,28,20,10,22,27)(14,23,15,24,31,30,32,26);(1,3)(2,8)(4,15)(5,18)(6,17)(7,16)(9,26)(10,24)(11,13)(12,29)(14,19)(20,32)(21,25)(22,31)(23,27)(28,30);(1,4)(2,9)(3,14)(5,22)(6,19)(7,20)(8,23)(10,11)(12,28)(13,30)(15,18)(16,31)(17,32)(21,27)(24,25)(26,29);(1,4)(2,10)(3,15)(5,19)(6,22)(7,20)(8,24)(9,11)(12,28)(13,26)(14,18)(16,32)(17,31)(21,27)(23,25)(29,30)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72241161, #72241171, #72241304, … |
| maximal normal subgroup of | #72241161, #72241171, #72241304, … |
| maximal quotient of | C2×#359619 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266815);