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 | |
3274 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)(9,10,13,18,15,17,14,12)(11,19,21,20,23,24,22,16);(2,4)(9,11,15,23)(10,16,17,20)(12,19,18,24)(13,22,14,21);(5,6,7,8)(9,10,13,18,15,17,14,12)(11,19,21,20,23,24,22,16);(9,12,14,17,15,18,13,10)(11,20,22,19,23,16,21,24);(10,17)(12,18)(16,20)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372865);