Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #32880 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4493 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)(7,10)(8,9)(11,12)(13,14)(15,22)(16,21)(17,19)(18,25)(20,23)(24,28)(26,27);(6,9)(10,12)(14,19)(20,27)(21,25)(22,28);(13,15)(14,20)(16,23)(17,24)(18,26)(19,27)(21,28)(22,25);(2,4)(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22);(5,7,8,11)(6,10,9,12)(15,24)(20,27)(22,28)(23,26)Polycyclic presentation:
2^9|d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72284628, #72310173, #72310179 |
| maximal normal subgroup of | #72284628, #72310173, #72310179 |
| maximal quotient of | #72310195, #72310223, #72318460 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536490);