Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | #35256 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,27)(17,22)(18,20)(19,25)(21,28)(23,26);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,17,25)(14,19,22,24)(16,26,28,18)(20,21,23,27);(1,3,4,2)(7,11)(9,12)(14,20)(15,25)(18,26)(19,27)(21,24)(22,23);(6,10)(9,12)(13,16)(14,21)(15,26)(17,28)(18,25)(19,23)(20,24)(22,27);(15,25)(16,28)(19,24)(21,27)Polycyclic presentation:
2^9|b^2=f,c^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340419);