Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,8,11,9,12,7,10)(13,14)(15,21,23,28)(16,20)(17,19)(18,22,26,24)(25,27);(6,10,12,11)(13,15)(14,18)(16,24)(17,23)(19,26)(20,28)(21,27)(22,25);(1,2,3,4)(6,10,12,11)(14,19)(16,25)(18,26)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di98 conjugacy classes — view the full character table.
| maximal subgroup of | #72227451, #72238860, C2×#110638, … |
| maximal normal subgroup of | #72238860, C2×#110638, #72239084, … |
| maximal quotient of | #72227430, C2×#110638, #72268626 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17834);