Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,17,19)(15,22,24,28)(16,21,18,26)(20,23,27,25);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,19)(20,27)(21,26)(22,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15,16,23,17,24,18,25)(14,20,26,28,19,27,21,22);(2,4)(5,8,9,7)(6,10,12,11)(13,16,17,18)(14,21,19,26)(15,23,24,25)(20,22,27,28);(15,24)(20,27)(22,28)(23,25)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351878);