Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | |
1867 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)(14,19)(15,23)(17,26)(20,24)(21,27)(22,28);(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,19,18,22,26,28)(15,24,23,16,21,20,27,25);(6,9)(10,11)(13,15,17,23,18,21,26,27)(14,20,19,25,22,24,28,16);(5,7,8,12)(6,10,9,11)(13,16,18,25)(14,21,22,15)(17,20,26,24)(19,27,28,23);(15,21)(16,25)(20,24)(23,27)Polycyclic presentation:
2^9|a^2=fgi,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=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371281);