Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2937 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)(9,10,13,18)(11,19,21,15)(12,17,14,23)(16,22,24,20);(1,2,3,4)(5,6)(7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,17)(18,19)(23,24);(6,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,17)(18,24)(19,23);(5,6)(7,8)(9,12,13,14)(10,17,18,23)(11,20,21,22)(15,24,19,16);(6,8)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72257706, #72257716, #72263857, … |
| maximal normal subgroup of | #72257706, #72257716, #72263857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481568);