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 | #33519 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
766 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)(6,11,14,26)(8,19)(9,23)(12,25,28,36)(13,24,20,15)(16,30)(17,34)(21,22)(27,35,31,29)(32,33);(5,6,9,24,10,14,23,15)(7,12,17,35,18,28,34,29)(8,20,19,11,22,13,21,26)(16,31,30,25,33,27,32,36);(2,4)(5,7,10,18)(6,12,14,28)(8,16,22,33)(9,17,23,34)(11,25,26,36)(13,27,20,31)(15,29,24,35)(19,30,21,32);(6,13)(8,21)(9,23)(11,24)(12,27)(14,20)(15,26)(16,32)(17,34)(19,22)(25,35)(28,31)(29,36)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,e^2=h,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 | #72250443, #72250863, #72250867, … |
| maximal normal subgroup of | #72250443, #72250863, #72250867, … |
| maximal quotient of | #72287109, #72287213, #72287416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405314);