Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1146 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)(11,16)(12,17)(13,18,22,15)(14,19)(20,24,21,23);(10,15)(13,22)(16,23)(17,24)(18,19)(20,21);(2,6)(7,8)(9,11,14,12)(10,16,19,17)(13,20,22,21)(15,23,18,24);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,19,16)(13,21,22,20)(15,24,18,23)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=ei,f^b=fgi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318204, #72318844, #72318858, … |
| maximal normal subgroup of | #72318204, #72318844, #72318858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387589);