Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,18,15,19,14,17)(11,20,21,16,23,24,22,12);(1,2)(3,4)(9,11,14,22,15,23,13,21)(10,16,18,24,19,12,17,20);(6,8);(2,4)(9,12)(10,11)(13,20)(14,24)(15,16)(17,22)(18,21)(19,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72309888, D4×#1765, #72310129 |
| maximal normal subgroup of | #72309888, D4×#1765, #72310129 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263296);