Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,10)(7,8)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,23)(24,27)(25,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,19,21,28)(16,25)(17,23)(18,24)(20,22,26,27);(6,11)(10,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22);(5,7,9,8)(6,10,11,12)(14,21)(19,28)(20,26)(22,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=i,e^2=h,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240819, #72249976, #72249980, … |
| maximal normal subgroup of | #72240819, #72249976, #72249980, … |
| maximal quotient of | #72240808, #72240809, #72257790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88950);