Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(11,21)(12,23)(13,14)(16,17)(18,19)(22,24);(1,2,4,3)(5,6)(7,8)(9,10,15,20)(11,22,23,17)(12,16,21,24)(13,19,14,18);(9,11)(10,16)(12,13)(14,21)(15,23)(17,19)(18,22)(20,24);(6,8)(9,12,15,21)(10,17,20,22)(11,13,23,14)(16,19,24,18);(1,3,4,2)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282037, #72282296, #72282518, … |
| maximal normal subgroup of | #72282037, #72282296, #72282518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426086);