Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(2,6)(3,4)(7,8)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21);(2,7)(3,4)(6,8);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72238774, #72240781, #72240785, … |
| maximal normal subgroup of | #72238774, #72240781, #72240785, … |
| maximal quotient of | #72257793, #72257835, #72285657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93706);