Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C26 | Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,18,14,16,13,17)(11,19,21,15,20,24,22,23);(1,3)(2,5,6,7)(4,8)(9,11,13,22,14,20,12,21)(10,15,18,24,16,23,17,19);(11,20)(15,23)(19,24)(21,22);(10,16)(15,23)(17,18)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=h,g^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 | #72235222, #72236746, #72236786, … |
| maximal normal subgroup of | #72235222, #72236746, #72236786, … |
| maximal quotient of | #72238897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88975);