Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,11,8,12)(13,14,18,21)(15,19,17,25)(16,23,24,27)(20,26,28,22);(1,2)(3,4)(6,10)(11,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(2,4)(13,16,18,24)(14,20,21,28)(15,22,17,26)(19,27,25,23);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(14,21)(19,25)(20,28)(23,27);(2,4)(5,8,9,7)(6,12,10,11)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485560);