Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | #23391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,22)(17,24)(18,20)(19,23)(21,26)(27,28);(1,2)(3,4)(6,9)(10,12)(14,20)(21,28)(22,25)(23,24);(2,4)(13,15,18,16)(14,21,20,28)(17,26,19,27)(22,23,25,24);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,22,20,25)(17,26,19,27)(21,23,28,24);(13,17,18,19)(14,23,20,24)(15,26,16,27)(21,25,28,22)Polycyclic presentation:
2^9|c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280687, #72283144, #72284948, … |
| maximal normal subgroup of | #72280687, #72283144, #72284948, … |
| maximal quotient of | C2×#6439342, #72311387, #72311569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346538);