Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3424 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)(9,10,14,19,15,17,13,18)(11,20,21,22,16,23,24,12);(6,7)(9,11)(10,12)(13,21)(14,24)(15,16)(17,22)(18,20)(19,23);(1,2,3,4)(5,6,8,7)(9,12)(10,16)(11,17)(13,23)(14,20)(15,22)(18,21)(19,24);(5,7,8,6)(10,17)(11,16)(18,19)(21,24);(9,13,15,14)(10,18,17,19)(11,21,16,24)(12,20,22,23)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72315230, #72315255, #72315259 |
| maximal normal subgroup of | #72315230, #72315255, #72315259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372766);