Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,19,21,18,22,17,27)(15,23,25,28,20,16,24,26);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,22,23)(17,24,19,25)(21,26,27,28);(6,9)(10,12)(13,16,18,23)(14,20,22,15)(17,26,19,28)(21,25,27,24);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,27)(15,24,20,25)(16,26,23,28);(14,22)(15,20)(21,27)(24,25)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342702);