Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2078 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)(13,14,18,21)(15,22,20,17)(16,24,25,23)(19,27,28,26);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15,17,14,18,20,22,21)(16,23,26,19,25,24,27,28);(6,9)(10,11)(13,16,17,26,18,25,22,27)(14,19,20,24,21,28,15,23);(5,7,8,12)(6,10,9,11)(16,25)(19,28)(23,24)(26,27);(13,17,18,22)(14,15,21,20)(16,26,25,27)(19,23,28,24)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=h,e^2=i,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^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371259);