Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110954 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1437 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,14,18,21)(12,15)(13,16)(17,23,24,20)(19,22);(2,6,8,7)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,23)(19,24);(2,6,8,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72227450, #72228200, #72232796, … |
| maximal normal subgroup of | #72228200, #72232796, #72233070, … |
| maximal quotient of | #72227430, #72227431, C2×#110998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18194);