Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,18,19,17,22,16,23)(15,21,24,28,26,20,25,27);(1,2)(3,4)(14,19)(15,24)(16,18)(20,21)(22,23)(25,26);(2,4)(5,7,8,12)(6,9,10,11)(13,15,18,24,17,26,16,25)(14,20,19,27,22,21,23,28);(2,4)(13,15,16,25,17,26,18,24)(14,21,23,27,22,20,19,28);(14,22)(19,23)(20,21)(27,28)Polycyclic presentation:
2^9|a^2=fg,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284383 |
| maximal normal subgroup of | #72284383 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353362);