Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,28)(24,26)(25,27);(2,4)(5,7,8,11)(6,9,10,12)(14,19)(15,23)(16,18)(20,27)(24,25)(26,28);(6,10)(9,12)(13,15)(14,20)(16,23)(17,24)(18,25)(19,26)(21,28)(22,27);(13,15)(14,20)(16,23)(17,24)(18,25)(19,26)(21,28)(22,27);(13,16,17,18)(14,21,19,22)(15,23,24,25)(20,28,26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281503, #72281724, #72281815, … |
| maximal normal subgroup of | #72281503, #72281724, #72281815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347706);