Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | |
1154 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)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,17)(18,20)(21,22,25,27,26,23,24,28);(1,2)(3,4)(6,10)(11,12)(14,17)(18,19)(21,23,25,28,26,22,24,27);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19)(21,22)(23,26)(24,27)(25,28);(5,8,9,7)(6,11,10,12)(14,17)(18,19)(21,24,26,25)(22,27,23,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=e,d^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280014);