Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,17,20,27)(8,14,23,30)(9,15,10,12)(13,24,28,18)(19,31,34,36)(21,29,22,25)(26,33,35,32);(2,4)(6,12)(7,18)(8,21)(9,10)(15,16)(17,27)(20,24)(22,23)(25,29)(26,31)(32,33)(35,36);(5,7)(6,13)(8,19)(9,18)(10,24)(11,20)(12,17)(14,26)(15,27)(16,28)(21,32)(22,33)(23,34)(25,31)(29,36)(30,35);(5,8)(6,14)(7,19)(9,21)(10,22)(11,23)(12,25)(13,26)(15,29)(16,30)(17,31)(18,32)(20,34)(24,33)(27,36)(28,35);(5,9,11,10)(6,12,16,15)(7,18,20,24)(8,21,23,22)(13,17,28,27)(14,25,30,29)(19,32,34,33)(26,31,35,36)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237505, #72286108, #72287020, … |
| maximal normal subgroup of | #72237505, #72286108, #72287020, … |
| maximal quotient of | #72310310, #72310312, C2×#6437611 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344807);