Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #18520 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,11,15,17,13,19)(12,16,21,18,22,23,20,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,18)(14,17)(15,19)(16,20)(21,23)(22,24);(1,3,5,4)(2,6,7,8)(9,10,15,17)(11,13,19,14)(12,16,22,23)(18,20,24,21);(2,7)(6,8)(9,12,14,21,15,22,13,20)(10,16,11,18,17,23,19,24)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,e^b=eh,f^b=fi,e^c=ei,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72287242 |
| maximal normal subgroup of | #72287242 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 117554);