Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1906 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)(13,14,18,21)(15,22,23,26)(16,25,20,17)(19,27,28,24);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15,17,24,18,23,25,27)(14,19,20,22,21,28,16,26);(6,9)(10,11)(13,16,17,14,18,20,25,21)(15,22,24,28,23,26,27,19);(5,7,8,12)(6,10,9,11)(15,23)(19,28)(22,26)(24,27);(13,17,18,25)(14,16,21,20)(15,24,23,27)(19,26,28,22)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=hi,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^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371267);