Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,15)(12,16,20,14)(13,17)(18,22,24,21)(19,23);(10,14)(12,20)(15,21)(16,17)(18,24)(22,23);(1,3,5,4)(2,6,8,7);(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,e^b=ei,f^b=fgi,f^e=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72338666, #72338671, #72338756, … |
| maximal normal subgroup of | #72338666, #72338671, #72338756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387300);