Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(14,18)(16,24)(19,27)(20,25)(21,28)(22,26);(5,7,9,8)(6,10,12,11)(13,14,17,20)(15,21,23,27)(16,25,24,18)(19,26,28,22);(6,11)(7,8)(10,12);(6,12)(10,11)(13,15,17,23)(14,19,20,28)(16,26,24,22)(18,27,25,21)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309438, #72309459, #72309507, … |
| maximal normal subgroup of | #72309438, #72309459, #72309507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128080);