Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #36644 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,18,22,17,20,16,21)(15,19,24,26,23,27,25,28);(5,7,11,6,9,12,8,10)(13,15,16,24,17,23,18,25)(14,19,22,28,20,27,21,26);(1,2,3,4)(6,10)(7,12)(14,20)(15,23)(21,22)(24,25);(13,16,17,18)(14,21,20,22)(15,24,23,25)(19,26,27,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111970);