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)(5,6,11,18,12,19,13,32)(7,20,23,33,24,36,26,15)(8,25,27,35,22,34,21,14)(9,28,10,31,29,17,30,16);(2,4)(5,7,12,24)(6,14)(8,9,22,29)(10,21,30,27)(11,23,13,26)(15,31)(16,33)(17,20)(18,25)(19,35)(28,36)(32,34);(5,8,11,27,12,22,13,21)(6,15,18,20,19,33,32,36)(7,9,23,10,24,29,26,30)(14,31,25,17,35,16,34,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=h,f^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 | #72239852, #72239855, #72239863, … |
| maximal normal subgroup of | #72239852, #72239855, #72239863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54467);