Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2656 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)(5,6,7,8)(9,10)(11,20)(12,23)(13,18)(14,19)(15,21)(16,17)(22,24);(1,2)(3,4)(9,11)(10,16)(12,21)(13,23)(14,17)(15,22)(18,24)(19,20);(9,12,15,16,14,24,13,20)(10,17,18,23,19,11,21,22);(2,4)(9,13,14,15)(10,18,19,21)(11,22,17,23)(12,20,24,16);(11,17)(12,24)(16,20)(22,23)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72284272, #72284404 |
| maximal normal subgroup of | #72284272, #72284404 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287277);