Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1728 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)(3,4)(6,7)(13,14)(15,26)(16,20)(17,24)(18,23)(19,25)(21,27)(22,28);(5,6,8,7)(9,10)(11,12)(13,15,17,16,19,28,18,27)(14,20,23,26,25,21,24,22);(2,4)(5,7,8,6)(10,12)(13,15,17,16,19,28,18,27)(14,21,23,22,25,20,24,26);(13,16,17,28,19,27,18,15)(14,22,23,20,25,26,24,21)Polycyclic presentation:
2^9|b^2=e,c^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264864, #72264877, #72264924, … |
| maximal normal subgroup of | #72264864, #72264877, #72264924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319363);