Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,22)(15,23,21,18)(16,26,25,24)(17,20,28,27);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15,18,14,19,21,23,22)(16,24,27,17,25,26,20,28);(6,9)(10,11)(13,16,18,27,19,25,23,20)(14,17,21,26,22,28,15,24);(5,7,8,12)(6,10,9,11)(13,17)(14,20)(15,25)(16,21)(18,26)(19,28)(22,27)(23,24);(16,25)(17,28)(20,27)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370992);