Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234325 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36614 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(5,8);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|b^2=h,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233812, #72233814, #72233817, … |
| maximal normal subgroup of | #72233812, #72233814, #72233817, … |
| maximal quotient of | #72316710, #72316732, #72316774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141522);