Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2967 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)(6,7)(10,14)(11,17)(16,22)(18,20);(1,2,4,3)(6,7)(11,18)(13,19)(15,23)(16,22)(17,20)(21,24);(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,21)(18,23)(19,22)(20,24);(5,7,8,6)(9,11)(10,15)(12,17)(13,20)(14,21)(16,24)(18,19)(22,23);(1,3,4,2)(10,14)(15,21)(16,22)(23,24)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72263890, #72263913, #72263951 |
| maximal normal subgroup of | #72263890, #72263913, #72263951 |
| maximal quotient of | #72264150, #72264329 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470315);