Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483839 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,12,15,16)(7,18,21,20)(8,13)(9,24)(10,23)(14,30)(17,28,26,27)(19,31)(22,29,34,35)(33,36);(1,2,3,4)(5,6,13,28,23,26,8,16,11,15,24,27,10,17,9,12)(7,19,35,32,34,14,18,33,21,30,29,25,22,31,20,36);(5,7,9,18,10,22,24,35,11,21,8,20,23,34,13,29)(6,14,16,32,17,19,28,36,15,31,12,25,26,30,27,33);(6,15)(12,16)(14,31)(17,26)(19,30)(25,32)(27,28)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224826, #72224936, #72226807, … |
| maximal normal subgroup of | #72224826, #72224936, #72226807, … |
| maximal quotient of | #72224819, #72319122, #72319127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402437);