Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483839 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
774 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)(5,6,11,15,12,17,28,16)(7,18,23,19,20,25,24,36)(8,10,26,30,14,9,27,29)(13,31,33,21,32,35,34,22);(5,7,10,21,11,24,30,31,12,20,9,22,28,23,29,35)(6,13,8,25,16,33,27,36,17,32,14,18,15,34,26,19);(2,4)(5,8,9,16,11,27,29,17,12,14,10,15,28,26,30,6)(7,19,21,32,24,25,31,34,20,36,22,13,23,18,35,33);(7,20)(13,32)(18,25)(19,36)(21,22)(23,24)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=h,g^2=i,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 | #72264797, #72264817, #72264853, … |
| maximal normal subgroup of | #72264797, #72264817, #72264853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402125);