Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33520 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,17,12,18,8,14)(7,19,20,34,22,35,28,33)(9,15,16,27,25,24,23,10)(13,29,30,21,32,26,31,36);(5,7,12,22)(6,13)(8,20,11,28)(9,19)(10,26,27,29)(14,30)(15,21,24,36)(16,33)(17,31)(18,32)(23,34)(25,35);(2,4)(5,8,12,11)(6,14,18,17)(7,20,22,28)(9,23,25,16)(10,24,27,15)(13,30,32,31)(19,34,35,33)(21,26,36,29);(5,9)(6,15)(7,13)(8,23)(10,14)(11,16)(12,25)(17,27)(18,24)(19,29)(20,30)(21,34)(22,32)(26,35)(28,31)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=eh,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250638, #72250863, #72250872, … |
| maximal normal subgroup of | #72250638, #72250863, #72250872, … |
| maximal quotient of | #72287294, #72287350, #72287499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405318);