Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,22,23,26,24,27,28,25);(1,3,4,2)(6,10)(11,12)(14,17)(15,19)(22,25)(23,28)(26,27);(5,7,9,8)(6,11,10,12)(13,15,16,19)(14,18,17,20)(22,25)(23,28)(26,27);(5,8,9,7)(6,11,10,12)(14,17)(18,20)(22,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=efi,d^2=efgi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280800);