Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13867 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,25)(3,14,15,9,16,28,26,27)(4,17,19,10,20,30,18,31)(5,21,22,11,24,29,23,32);(1,3)(2,9)(4,5)(6,26)(7,15)(8,16)(10,29)(11,17)(12,28)(13,27)(14,25)(18,23)(19,22)(20,24)(21,31)(30,32);(1,4)(2,10)(3,5)(6,18)(7,19)(8,20)(9,21)(11,28)(12,30)(13,31)(14,32)(15,22)(16,24)(17,25)(23,26)(27,29);(1,5)(2,11)(3,4)(6,23)(7,22)(8,24)(9,17)(10,28)(12,29)(13,32)(14,31)(15,19)(16,20)(18,26)(21,25)(27,30)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238012, #72241171, #72241172, … |
| maximal normal subgroup of | #72238012, #72241171, #72241172, … |
| maximal quotient of | #72263005, C2×#359673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266869);