Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17,13,19,12,18)(7,20,8,25,24,35,23,32)(9,16,10,15,28,33,27,34)(14,26,30,29,31,22,36,21);(2,4)(5,7,12,23,13,24,11,8)(6,14,17,30,19,31,18,36)(9,26,27,21,28,22,10,29)(15,32,33,20,34,25,16,35);(5,8,11,24,13,23,12,7)(6,14,17,30,19,31,18,36)(9,21,10,26,28,29,27,22)(15,25,33,35,34,32,16,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72322024, #72322036, #72322061, … |
| maximal normal subgroup of | #72322024, #72322036, #72322061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54468);