Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,15,14,17,12,18)(11,16,21,22,19,23,20,24);(1,2,3,4)(10,15)(12,13)(16,22)(17,18)(20,21)(23,24);(6,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24);(11,19)(16,23)(20,21)(22,24);(10,17)(15,18)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^b=fi,h^b=hi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305326, #72305335, #72305342, … |
| maximal normal subgroup of | #72305326, #72305335, #72305342, … |
| maximal quotient of | #72306026, #72310244, #72318201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278581);