Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10,13,19,14,18,15,16)(11,20,22,23,12,17,21,24);(2,4)(10,16)(11,21)(12,22)(13,15)(17,20)(18,19);(5,6,7,8)(9,11,15,21,14,12,13,22)(10,17,16,23,18,20,19,24);(9,12,15,22,14,11,13,21)(10,17,16,23,18,20,19,24);(10,18)(11,12)(16,19)(21,22)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282937, #72283132, #72283296, … |
| maximal normal subgroup of | #72282937, #72283132, #72283296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372864);