Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,8)(9,10,11,12)(13,14)(15,23)(16,21)(17,20)(18,22)(19,26)(24,27)(25,28);(1,3,4,2)(10,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,23)(21,27)(22,28);(2,3)(9,10)(11,12)(14,20)(15,24)(16,18)(19,27)(23,28)(25,26);(6,8)(10,12)(13,16,17,18)(14,21,20,22)(15,25,26,24)(19,27,23,28)Polycyclic presentation:
2^9|a^2=fi,b^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363743);