Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392727 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12)(3,7,11,6)(4,15)(5,14)(9,18)(10,19)(13,17)(16,29,31,27)(20,25)(21,26)(22,24)(23,32,28,30);(1,3,6,2,8,11,7,12)(4,13,22,9,18,24,17,15)(5,14,20,10,21,26,19,25)(16,27,30,23,31,29,32,28);(1,4)(2,9)(3,13)(5,16)(6,22)(7,17)(8,18)(10,23)(11,24)(12,15)(14,27)(19,32)(20,30)(21,31)(25,28)(26,29);(1,5)(2,10)(3,14)(4,16)(6,20)(7,19)(8,21)(9,23)(11,26)(12,25)(13,27)(15,28)(17,32)(18,31)(22,30)(24,29)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72219682, #72219687, … |
| maximal normal subgroup of | #72215931, #72219682, #72219687, … |
| maximal quotient of | C2×#392728, #72316824 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299924);