Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,15)(11,13)(12,19)(16,24)(17,23)(18,21)(20,22);(9,10)(11,16)(12,20)(13,22)(14,17)(15,19)(18,23)(21,24);(2,5)(6,8);(9,11,14,13)(10,16,17,22)(12,18,21,19)(15,20,23,24)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,e^b=eg,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72260397, #72260401, #72260414, … |
| maximal normal subgroup of | #72260397, #72260401, #72260414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388463);