Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433040 |
| 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 | #36614 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(5,7)(6,9)(8,11)(10,12)(13,15,16,23,17,22,25,26)(14,18,19,21,20,27,24,28);(2,4)(7,11)(9,12);(6,10)(9,12);(15,22)(18,27)(21,28)(23,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^b=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281503, #72281793, #72283829, … |
| maximal normal subgroup of | #72281503, #72281793, #72283829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340237);