Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3066 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)(9,10,13,19,14,20,15,18)(11,21,22,17,23,24,12,16);(5,6)(7,8)(9,11)(10,16)(12,13)(14,23)(15,22)(17,20)(18,21)(19,24);(1,2,3,4)(6,8)(9,12,13,11,14,22,15,23)(10,17,19,24,20,16,18,21);(6,8)(9,12,13,11,14,22,15,23)(10,16,19,21,20,17,18,24);(9,13,14,15)(10,18,20,19)(11,12,23,22)(16,21,17,24)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,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 | #72281103, #72281161, #72281198, … |
| maximal normal subgroup of | #72281103, #72281161, #72281198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351587);