Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,19,23,21);(9,11,13,21,14,20,12,19)(10,15,16,18,17,23,22,24);(2,6)(7,8);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ci,g^a=gh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224691, #72224692, #72224694, … |
| maximal normal subgroup of | #72224691, #72224692, #72224694, … |
| maximal quotient of | #72257629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88076);